diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmkjv" "b/data_all_eng_slimpj/shuffled/split2/finalzzmkjv"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmkjv"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nRecently, several intriguing threads relating derived categories and\narithmetic geometry have emerged and motivated general structure questions \nfor $k$-linear triangulated categories for arbitrary\nfields $k$.  Such exploration has yielded many nice applications as well as \nfurther enticing problems, see as a sampling \\cite{AKW, AAGZ, ADPZ, HT, Honigs, LiebMaulSnow}. \nMeanwhile over $\\mathbb{C}$, structural results for derived\ncategories seem to have deep implications for the underlying birational geometry, e.g.\n\\cite{AT, ABB, BB, BMMSS, KuznetsovCubic4fold, Vial}. Taking these together,\nderived categories become an important invariant for studying birational\ngeometry over a general field \\cite{AB}. A further benefit of this\nnoncommutative approach is direct utility for solving problems in algebraic $K$-theory,\nfor example \\cite{MerkPan}.\n\nWith such tantalizing ties, one would like a fertile testing ground for\nquestions. In this paper, we begin a systematic study of one such area:\nderived categories of arithmetic toric varieties. This area has the\nfollowing nice features:\n\n\\begin{itemize}\n \\item rationality issues are deep in general but tractable in examples,\n \\item robust tools already exist to investigate derived categories over\nthe separable closure,\n \\item and specific questions are often amenable to computational\nexperimentation.\n\\end{itemize}\n\nOne of the best tools for understanding a derived category is an\n\\emph{exceptional collection} consisting of \\emph{exceptional objects}.\nAs originally conceived in \\cite{Beilinson}, an exceptional object of a $k$-linear derived category is one whose endomorphism algebra\nis isomorphic to the base field $k$.\nWhen $k$ is not algebraically closed, this definition is too restrictive and\ninstead we use the existing notion: an object of $\\mathsf{D^b}(X)$ is \\emph{exceptional} if its endomorphism algebra is a division algebra. \nDetails are discussed in Section~\\ref{sect:galdesc} below.\n\nWe illustrate this more general notion for two arithmetic toric\nvarieties.\nThe real conic $X = \\{x^2 + y^2 + z^2 = 0\\} \\subset \\mathbb{P}^2_{{\\mathbb R}}$\nhas an exceptional collection $\\{ \\O, \\mathcal{F} \\}$, where\n$\\text{End}(\\mathcal{F})$ is isomorphic to the quaternion algebra\n$\\mathbb{H}$.\nOver ${\\mathbb C}$, we have $X_{\\mathbb C} \\simeq {\\mathbb P}^1_{\\mathbb C}$ and $\\mathcal{F}\\otimes_{{\\mathbb R}} {\\mathbb C} \\simeq \\O(1)^{\\oplus 2}$.\nAs another example, consider the Weil restriction $Y$ of ${\\mathbb P}^1_{\\mathbb C}$ over\n${\\mathbb R}$ (``${\\mathbb P}^1({\\mathbb C})$ viewed as an ${\\mathbb R}$-variety'').\nHere $Y$ has an exceptional collection $\\{ \\O, \\mathcal{G},\n\\mathcal{H}\\}$ where $\\text{End}(\\mathcal{G}) \\simeq {\\mathbb C}$ and\n$\\text{End}(\\mathcal{H})\n\\simeq {\\mathbb R}$.\nOver ${\\mathbb C}$, we have $Y\\otimes_{{\\mathbb R}} {\\mathbb C} \\simeq {\\mathbb P}^1 \\times {\\mathbb P}^1$\nwith\n$\\mathcal{G} \\otimes_{{\\mathbb R}} {\\mathbb C}\n\\simeq \\O(1,0) \\oplus \\O(0,1)$ and $\\mathcal{H} \\otimes_{{\\mathbb R}} {\\mathbb C}\\simeq  \\O(1,1)$,\nwhere $\\O(i, j) = \\pi_1^*\\O(i) \\otimes \\pi_2 ^*\\O(j)$.\n\nA central question for derived categories of arithmetic toric varieties is the following: \n\n\\begin{question}\\label{quest:main}\n Let $X$ be a smooth projective toric variety over an arbitrary field.  Does $X$ admit a full exceptional collection? If so, does $X$ possess a full exceptional collection of sheaves?\n\\end{question}\n \nOver an algebraically-closed field of characteristic zero, there is always a full exceptional collection of objects \\cite{Kawamata,Kawamata2} while the question of a full exceptional collection of sheaves is due to Orlov. \nMaking allowances for different language,\nthe question is also known to have a positive answer for\nSeveri-Brauer varieties~\\cite{AB,Bernardara},\nminimal toric surfaces~\\cite{BSS},\nand smooth projective toric varieties with absolute Picard rank at most\n$2$~\\cite{Yan}.\n\nIn this article, we provide further evidence for a positive answer to\nQuestion~\\ref{quest:main},\ntreating cases with low dimension or a high degree of symmetry.\n\n\\begin{thm} \\label{thm:examples}\n The following possess full exceptional collections of sheaves:\n \\begin{itemize}\n  \\item smooth toric surfaces (Proposition~\\ref{prop:surface}),\n  \\item smooth toric Fano 3-folds (Proposition~\\ref{prop:3fold}), \n  \\item all forms of 43 of the 124 smooth split toric Fano 4-folds (Section~\\ref{sect:4fold}),\n  \\item all forms of centrally symmetric toric Fano varieties (Corollary~\\ref{cor:centsym}), and\n  \\item and all forms in characteristic zero of toric varieties corresponding to Weyl fans of root systems of type $A$ (Proposition~\\ref{prop:X(An)excpcoll}).\n \\end{itemize}\n\\end{thm}\n\nOur results leverage extant work in the algebraically closed case\nsuch as \\cite{Uehara} for 3-folds and\\cite{Prabhu} for 4-folds. We use Castravet and Tevelev's recently discovered exceptional collection for $X(A_n)$ \\cite{CT}. \nFor the centrally symmetric toric Fano varieties (which are\nproducts of ``generalized del Pezzo varieties'' and projective lines\n\\cite{VosKly}), we use an explicit exceptional\ncollection (see also \\cite{BDMdP}) closely related to the one found in \\cite{CT}.\nUp to a twist by a line bundle, the authors had independently\ndiscovered the exact same collection!\nThis suggests that symmetry imposes strong conditions on the\npossible exceptional collections, which paradoxically makes them easier to find.\n\nTo study arithmetic exceptional collections, we establish\nan effective Galois descent result for general exceptional collections.\nThis applies to general varieties, not just toric ones.\n\n\\begin{thm}[Theorem~\\ref{thm:descblocks}, Lemma~\\ref{lem:Galconverse}]\n Let $X$ be a $k$-scheme and $L\/k$ a $G$-Galois extension.  Then $X_L$ admits a full (resp. strong) $G$-stable exceptional collection of objects of $\\mathsf{D^b}(X_L)$ (resp. sheaves, resp. vector bundles) if and only if $X$ admits a full (resp. strong) exceptional collection of objects of $\\mathsf{D^b}(X)$ (resp. sheaves, resp. vector bundles).\n\\end{thm}\n\nWe highlight one corollary of a positive answer to\nQuestion~\\ref{quest:main}. Arithmetic toric varieties are also studied\nin \\cite{MerkPan}, which focused on computing their algebraic $K$-groups\nvia decompositions in a certain noncommutative motivic category of\n$K_0$-correspondences. They showed that for an arithmetic toric\n$k$-variety $X$ with $G = \\text{Gal}(k^s\/k)$, the group $K_0(X_{k^s})$\nis a direct summand of a \\emph{permutation $G$-module} (there exists a\n${\\mathbb Z}$-basis permuted by $G$).\n\n\\begin{question}[Merkurjev-Panin \\cite{MerkPan}]\\label{quest:MP}\nLet $X$ be an arithmetic toric variety over $k$ and $G = \\operatorname{Gal}(k^s\/k)$.  Is $K_0(X_{k^s})$ always a permutation $G$-module?\n\\end{question}\n\nQuestion~\\ref{quest:main} can be viewed as a categorification of Question~\\ref{quest:MP} as any such exceptional collection over $k$ immediately gives a permutation basis. \n\nIn order to show that every toric variety has a full exceptional collection\nover ${\\mathbb C}$, the main tool used in \\cite{Kawamata,Kawamata2} was the\nminimal model program (MMP) in birational geometry.\nThe basic building blocks are toric stacks with Picard rank one,\nwhich always have full strong exceptional collections of line bundles.\nIndeed, runs of the MMP can be leveraged to effectively produce\nexceptional collections \\cite{BFK}.\n\nOver a non-closed field, one hopes to use the Galois-equivariant MMP,\nbut the situation is more complicated.\nThe most basic building blocks in this framework are those varieties $X$\nwhich have $\\rho^G = \\text{rank} (\\text{Pic}(X)^G )= 1$.\nBased on the results above and the hope of using the MMP in the\narithmetic situation, we ask the following\nquestion in the vein of \\cite{King,BH,CM-RFrob}:\n\n\\begin{question}\\label{quest:invpicrank1}\nLet $X$ be a smooth toric $k$-variety and $L\/k$ a $G$-Galois splitting\nfield.  If $\\operatorname{Pic}(X_L)^G$ is of rank 1, does $X_L$ possess\na full strong $G$-stable exceptional collection consisting of line\nbundles?\n\\end{question}\n\n\\subsection*{Acknowledgements}\n\nThe first author was partially supported by NSF DMS-1501813. He would\nalso like to thank the Institute for Advanced Study for providing a\nwonderful research environment. These ideas were developed during his\nmembership.  The first author benefited from discussions with Alicia\nLamarche.\nThe second author was partially supported by NSA grant\nH98230-16-1-0309.\nThe third author would like to thank the Hausdorff Institute for their\nhospitality and lively research environment during the \\emph{K-theory\nand related fields} trimester program. A large portion of this\nmanuscript was drafted during his time in Bonn.\nAll authors also thank Fei Xie for pointing out that, due to an editing\nerror, in a previous version of this paper,\nProposition~\\ref{prop:surface} stated that all smooth toric surfaces\nhave strong collections of vector bundles instead of a not-necessarily\nstrong collection of sheaves as was proven.\nThe authors would also like to thank an anonymous referee for useful\ncomments.\n\n\n\\subsection*{Organization}\n\nSection~\\ref{sect:galdesc} treats Galois descent of exceptional\ncollections consisting of objects on (possibly non-toric) varieties. In\nSection~\\ref{section:toric}, we recall appropriate definitions of\narithmetic toric varieties and establish additional descent results\nwhich are specific to toric varieties.  In\nSection~\\ref{section:minimal}, we consider a range of examples.  We begin by treating toric surfaces, followed by toric Fano 3-folds.  For toric Fano 4-folds, we give partial results. We conclude by investigating the class of centrally symmetric toric Fano varieties, including the generalized del Pezzo varieties, and handling toric varieties associated to root systems of type $A$. \n\n\n\\subsection*{Notation} Throughout, $k$ denotes an arbitrary field and\n$k^s$ a separable closure. A \\emph{variety} is a geometrically integral\nseparated scheme of finite type over $k$. All our schemes will be\nquasi-compact and quasi-separated. For a $k$-scheme $X$ and field\nextension $L\/k$, we write $X_L : = X \\times _{\\operatorname{Spec} k} \\operatorname{Spec} L$.  If $A$\nis a $k$-algebra, we write $A_L = A\\otimes _{k} L$.\nWe use $\\mathsf{D^b}(X)$ to denote the bounded derived category\n$\\mathsf{D^b}(\\text{Coh}(X))$. For an $\\O_X$-algebra $A$, we write\n$\\mathsf{D^b}(A)$ for the bounded derived category of complexes of\n$A$-modules which are coherent $\\O_X$-modules.\n\n\n\\section{Galois descent and exceptional collections}\\label{sect:galdesc}\n\nIn this section we develop Galois descent for exceptional collections (in a generalized sense). We begin by recalling some definitions and conventions concerning structure theory of derived categories of schemes.  We then give our main descent results for $G$-stable exceptional collections (Theorem~\\ref{thm:descblocks}). We complete the section by collecting a few useful consequences to be used in the sequel.\n\n\n\\subsection{Exceptional collections}\n\nWe give some conventions for semiorthogonal decompositions of derived categories and in particular exceptional collections.  Such collections have been widely studied over algebraically closed fields but have recently been treated in more generality \\cite{AAGZ, AB, ABB, Bernardara, BSS, Elagin, Xie, Yan}.  We refer the reader to Remarks~\\ref{rem:descSOD} and \\ref{rem:elagin} for added details on some of these results.  \n\nFor a triangulated category $\\mathsf{T}$, we use the standard notation $\\text{Ext}^n_{\\mathsf{T}}(A, B) = \\hom _{\\mathsf{T}} (A, B[n])$.  For objects $A, B $ of $\\mathsf{D^b}(X)$, we use  $\\text{End}_X(A)$ and $\\text{Ext}_X^n(A, B)$ to denote $\\text{End}_{\\mathsf{D^b}(X)}(A)$ and $\\text{Ext}^n_{\\mathsf{D^b}(X)}(A, B)$, respectively.\n\n\\begin{defn}[see \\cite{BK}]\nLet $\\mathsf{T}$ be a triangulated category.  A full triangulated subcategory of $\\mathsf{T}$ is \\emph{admissible} if its inclusion functor admits left and right adjoints.  A \\emph{semiorthogonal decomposition} of $\\mathsf{T}$ is a sequence of admissible subcategories $\\mathsf{C}_1, ..., \\mathsf{C}_s$ such that \n\\begin{enumerate}\n\\item $\\hom _{\\mathsf{T}}(A_i, A_j) = 0$ for all $A_i \\in \\operatorname{Ob} (\\mathsf{C}_i)$, $A_j \\in \\operatorname{Ob} (\\mathsf{C}_j)$ whenever $i > j$.\n\\item For each object $T$ of $\\mathsf{T},$ there is a sequence of morphisms $0 = T_s \\to \\cdots \\to T_0 = T$ such that the cone of $T_i \\to T_{i-1}$ is an object of $\\mathsf{C}_i$ for all $i = 1,..., s$.\n\\end{enumerate}\nWe use  $\\mathsf{T} = \\langle \\mathsf{C}_1,..., \\mathsf{C}_s\\rangle$ to denote such a decomposition.\n\\end{defn}\n\nParticularly nice examples of semiorthogonal decompositions are given by exceptional collections, the study of which goes back to Beilinson \\cite{Beilinson}.\n\n\\begin{defn}\\label{def:exceptional}\nLet $\\mathsf{T}$ be a $k$-linear triangulated category.  An object $E$ in $\\mathsf{T}$ is \\emph{exceptional} if the following conditions hold:\n\\begin{enumerate}\n\\item $\\text{End}_{\\mathsf{T}}(E)$ is a division $k$-algebra.\n\\item $\\text{Ext}^n_{\\mathsf{T}}(E, E) = 0$ for $n \\neq 0$.\n\\end{enumerate}\nA totally ordered set $\\mathsf{E} = \\{E_1, ..., E_s\\}$ of exceptional objects is an \\emph{exceptional collection} if $\\text{Ext}^n_{\\mathsf{T}}(E_i, E_j) = 0$ for all integers $n$ whenever $i >j$.  An exceptional collection is $\\emph{full}$ if it generates $\\mathsf{T}$, i.e., the smallest thick subcategory of $\\mathsf{T}$ containing $\\mathsf{E}$ is all of $\\mathsf{D^b}(X)$.  An exceptional collection is \\emph{strong} if $\\text{Ext}^n_{\\mathsf{T}}(E_i, E_j) = 0$ whenever $n \\neq 0$.  An \\emph{exceptional block} is an exceptional collection $\\mathsf{E} = \\{ E_1, ..., E_s\\}$ such that $\\text{Ext}^n_{\\mathsf{T}}(E_i, E_j) = 0$ for every $n$ whenever $i \\neq j$. Given an exceptional collection $\\mathsf{E} = \\{E_1, ..., E_s\\}$, we denote by $\\langle \\mathsf{E} \\rangle$ the category generated by the objects $E_i$.\n\\end{defn}\n\n\\begin{rem}\nOur notion of exceptional object generalizes the classical one, where item $(1)$ of Definition~\\ref{def:exceptional} is replaced by:  $\\text{End}_{\\mathsf{T}}(E) = k$ \\cite[$\\S$2]{Bondal}.  Over algebraically or separably closed fields, these definitions agree.  Over non-closed fields, the classical definition is too restrictive to allow for the use of interesting arithmetic invariants in the study of exceptional collections on twisted forms, e.g., Brauer classes.\n\\end{rem}\n\n\\begin{prop}[Thm. 3.2 \\cite{Bondal}]\\label{prop:exctosod}\nLet $X$ be a $k$-scheme with exceptional collection $\\{E_1,..., E_s\\}$.  If $\\mathscr{E}_i $ is the category generated by $E_i$, there is a semiorthogonal decomposition $\\mathsf{D^b}(X) = \\langle \\mathscr{E}_1,..., \\mathscr{E}_s, \\mathsf{A} \\rangle$, where $\\mathsf{A}$ is the full subcategory with objects $A$ such that $\\operatorname{Hom}_X(A, E_i) = 0$ for all $i$.\n\\end{prop}\n\n\\begin{rem}\n The reference assumes smoothness and projectivity but the conclusion is independent of this. Note further that if $X$ admits a full exceptional collection then it is automatically smooth and proper by \\cite[Propositions 3.30 and 3.31]{OrlovNCstuff}.\n\\end{rem}\n\nThe existence of an exceptional collection on a scheme $X$ provides a means of studying derived geometry of $X$ in purely algebraic terms.  Indeed, in such a situation, one may identify an ``underlying\" $k$-algebra which is derived equivalent to $X$.  For exceptional blocks, one obtains a similar but slightly stronger fact.\n\n\\begin{prop}[Thm. 6.2 \\cite{Bondal}]\\label{thm:tilting}\nLet $X$ be a smooth projective $k$-scheme and let $\\{E_1,..., E_n\\}$ be a full strong exceptional collection on $\\mathsf{D^b}(X)$.  Let $\\mathcal{E}  = \\bigoplus E_i$ and $A = \\operatorname{End}(\\mathcal{E})$.  Then $\\mathsf{R} \\hom _{\\mathsf{D^b}(X)}(\\mathcal{E}, -) : \\mathsf{D^b}(X) \\to \\mathsf{D^b}(A)$ is a $k$-linear equivalence.\n\\end{prop}\n\n\\begin{prop}\nIf $\\mathsf{E} = \\{E_1,..., E_s\\}$ is an exceptional block with $\\operatorname{End}(E_i) = D_i$, there is a $k$-algebra isomorphism $\\operatorname{End}(\\bigoplus E_i) \\simeq D_1 \\times \\cdots \\times D_s$, and hence a $k$-linear equivalence $\\langle \\mathsf{E} \\rangle \\simeq  \\mathsf{D^b}(D_1 \\times \\cdots \\times D_n)$.\n\\end{prop}\n\nThe object $\\mathcal{E} = \\oplus E_i$ of Proposition~\\ref{thm:tilting} is usually called a \\emph{tilting object}.  If each $E_i$ is a sheaf (resp. vector bundle), then $E$ is called a \\emph{tilting sheaf} (resp. \\emph{tilting bundle}).  Until recently, the theory of tilting objects has served as the main tool for extending the study of exceptional collections to non-closed fields.  The results above show that any exceptional collection gives rise to both a tilting object and a semiorthogonal decomposition, and thus the admission of such a collection is a particularly special property of a given triangulated category.  Our aim in the following subsection is to extend descent results for semiorthogonal decompositions and tilting objects to (our more general notion of) exceptional collections. We give a formal definition of tilting object for completeness.\n\n\\begin{defn}\nA \\emph{tilting object} for a $k$-scheme $X$ is an object $\\mathcal{E}$ of $\\mathsf{D^b}(X)$ which satisfies the following conditions:\n\\begin{enumerate}\n\\item $\\text{Ext}_X ^n (\\mathcal{E}, \\mathcal{E}) = 0$ for $n > 0$.\n\n\\item $\\mathcal{E}$ generates $\\mathsf{D^b}(X)$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{rem}[$K$-theory and motivic decompositions]\n\nExceptional collections have a particularly interesting manifestation in the realm of noncommutative motives.  Indeed, an exceptional collection $\\{E_1,..., E_s\\}$ on a smooth projective variety $X$ yields a decomposition $U(X) \\simeq \\bigoplus _i U(D_i)$ of its corresponding universal additive invariant \\cite[$\\S$2.3]{Tabuada}, where $D_i = \\text{End}(E_i)$. This defines a motivic decomposition by viewing $X$ as an object in the Merkurjev-Panin category of $K$-motives \\cite{MerkPan} or Kontsevich's category of noncommutative Chow motives \\cite[Thm. 6.10]{Tabuada2} via its associated dg-category of perfect complexes.\n\nOne nice consequence is that this decomposition is detected by algebraic $K$-groups \\cite[Prop. 1.10]{AB} in addition to a slew of other additive invariants in the sense of Tabuada \\cite[$\\S$2.2]{Tabuada}. Such invariants include algebraic $K$-theory with coefficients, homotopy $K$-theory, \\'{e}tale $K$-theory, (topological) Hochschild homology, and (topological) cyclic homology.\n\\end{rem}\n\n\n\\subsection{Galois descent} We develop Galois descent for exceptional\ncollections consisting of objects in the derived category\n$\\mathsf{D^b}(X)$ of a (smooth projective) variety $X$.  Throughout this\nsection, pushforward and pullback functors are understood to be derived.\nFor a $k$-scheme $X$ and finite Galois extension $L\/k$, any element $g\n\\in \\text{Gal}(L\/k)$ defines a morphism of $k$-schemes $g: X_L \\to X_L$\nwhich in turn defines the functor $g^*: \\mathsf{D^b}(X_L) \\to \\mathsf{D^b}(X_L)$.\n\n\\begin{defn}\nLet $X$ be a scheme with an action of a group $G$.  A $G$-\\emph{stable\nexceptional collection} on $X$ is an exceptional collection $\\mathsf{E}\n= \\{E_1, ..., E_s\\}$ of objects in $\\mathsf{D^b}(X)$ such that\nfor all $g \\in G$ and $1 \\leq i \\leq s$ there exists $E \\in \\mathsf{E}$\nsuch that $g^*E_i \\simeq E$.\nWe say a $G$-stable exceptional collection $\\mathsf{E}$\nis a \\emph{$G$-orbit} if, for every pair of objects\n$E,E' \\in \\mathsf{E}$, there exists a $g \\in G$ such that\n$g^*E \\simeq E'$.\n\\end{defn}\n\n\\begin{rem}\\label{rem:invariant}\nA simple example of a $G$-stable exceptional collection is a\n$G$-\\emph{invariant} exceptional collection, i.e., an exceptional\ncollection $\\{E_1,..., E_s\\}$ such that $g^*E_i \\simeq E_i$ for all $1 \\leq i \\leq s$.  It is often the case that toric varieties admit exceptional collections consisting of line bundles.  If it is also the case that a group $G$ acts trivially on $\\text{Pic}(X)$, such a collection is automatically $G$-invariant, and hence $G$-stable (see Lemma~\\ref{lem:picinv}).\n\\end{rem}\n\n\\begin{lem}\\label{lem:collectiontoblocks}\nAny $G$-stable exceptional collection may be written as a collection of $G$-stable exceptional blocks (after possibly reordering).\n\\end{lem}\n\n\\begin{proof}\nThe decomposition of a $G$-stable exceptional collection into its $G$-orbits gives the desired exceptional blocks.  Let $\\mathsf{E}$ be a $G$-stable exceptional collection and for elements $E, E' \\in \\mathsf{E}$, we write $E \\leadsto E'$ if $\\text{Ext}^n(E, E') \\neq 0$ for some $n$.\n\n Let $\\mathsf{A} \\subset \\mathsf{E}$ be a $G$-orbit. To see that\n$\\mathsf{A}$ is an exceptional block, suppose that $E  \\leadsto E'$ for\n$E, E' \\in \\mathsf{A}$. Since $\\mathsf{A}$ is a $G$-orbit, $E' \\simeq g^*\nE$ for some $g \\in G$.  Thus, $E  \\leadsto g^* E$, and acting again by\n$g$, we have $g^*E  \\leadsto (g^2)^*E$. Since $A$ is finite, we have $E\n\\leadsto g^*E  \\leadsto \\cdots  \\leadsto (g^s)^*E  \\leadsto E$ for some\npositive integer $s$.  Thus,\nthere is no ordering of the elements of $\\mathsf{A}$\nsuch that they form a subset of an exceptional collection --- a\ncontradiction.\n\nIf $\\mathsf{B}$ is another $G$-orbit (distinct from $\\mathsf{A}$), we\nwould like to see that these blocks can be ordered to form an\nexceptional collection. We claim that for any $E \\in \\mathsf{A}$ and $F\n\\in \\mathsf{B}$, one has $E  \\leadsto F$ only if \n$\\mathsf{A}$ precedes\n$\\mathsf{B}$ in the collection $\\mathsf{E}$ (i.e., $\\text{Ext}^n(B, A) =\n0$ for all $n$ and all $A \\in \\mathsf{A}$, $B \\in \\mathsf{B}$).\nTo see this, assume that $E  \\leadsto\nF$ and $F  \\leadsto E'$ for some $E' \\in \\mathsf{A}.$ As $\\mathsf{A}$ is\na $G$-orbit, $E' \\simeq g^*E$ for some $g \\in G$. Hence, just as above,\nwe have a sequence $E  \\leadsto F  \\leadsto g^*E \\leadsto g^* F\n\\leadsto \\cdots  \\leadsto (g^s)^* F  \\leadsto E.$  Thus, there is no\nordering of the elements of $\\mathsf{A}$ and $\\mathsf{B}$ which forms an\nexceptional collection, contradicting the exceptionality of\n$\\mathsf{E}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:galconj}\nLet $X$ be a Noetherian $k$-scheme, $L\/k$ a finite Galois extension with group $G$, and $\\pi: X_L \\to X$ the natural projection map.  For any object $M $ in $\\mathsf{D^b}(X_L)$ there is a natural isomorphism $\\displaystyle \\pi^* \\pi_*( M) \\simeq \\bigoplus _{g\\in G} g^*M.$  \n\\end{lem}\n\n\\begin{proof}\n\nAs $\\pi$ is flat and affine, every coherent sheaf on $X$ is acyclic for $\\pi^\\ast$ and every coherent sheaf on $X_L$ is acyclic for $\\pi_\\ast$. Hence, the derived functors coincide with the application of $\\pi^\\ast$ or $\\pi_\\ast$ component-wise to a complex. Thus, it suffices to establish a natural isomorphism at the level of coherent sheaves. \n\nFor any object $M$ of $\\text{Coh}(X_L)$, we have $ \\pi_*M \\simeq \\pi_*\ng^* M $, and adjunction yields a natural transformation $ \\pi^* \\pi_*\n\\to g^*$.  Summing over all $g \\in G$ provides the transformation\n$\\alpha:   \\pi^* \\pi_* \\to \\oplus g^* $. We show this is an\nisomorphism.\n\nIt suffices to check that $\\alpha$ is an isomorphism on any affine patch, $\\operatorname{Spec} R$, of $X$. Passing to modules, we abuse notation and let $M$ be a finitely-generated module over $R_L = R \\otimes_k L$. Choose a presentation of $M$\n\\begin{displaymath}\n R_L^{\\oplus m} \\to R_L^{\\oplus n} \\to M \\to 0\n\\end{displaymath}\nand evaluate $\\alpha$ on the sequence to get the commutative diagram \n\\begin{center}\n \\begin{tikzpicture}\n  \\node at (-4,0.75) (pr1) {$R^{\\oplus m} \\otimes_k \\left( L \\otimes_k L \\right)$};\n  \\node at (-4,-0.75) (gr1) {$R^{\\oplus m} \\otimes_k \\left( \\oplus_g \\Gamma_g(L) \\right)$};\n  \\node at (0,0.75) (pr0) {$R^{\\oplus n} \\otimes_k \\left( L \\otimes_k L\\right) $};\n  \\node at (0,-0.75) (gr0) {$R^{\\oplus m} \\otimes_k \\left( \\oplus_g \\Gamma_g(L)\\right) $};\n  \\node at (3,0.75) (pm) {$M \\otimes_R R_L$};\n  \\node at (3,-0.75) (gm) {$\\oplus_g g^\\ast M$};\n  \\node at (5,0.75) (p0) {$0$};\n  \\node at (5,-0.75) (g0) {$0$};\n  \\draw[->] (pr1) -- (pr0);\n  \\draw[->] (pr0) -- (pm);\n  \\draw[->] (pm) -- (p0);\n  \\draw[->] (gm) -- (g0);\n  \\draw[->] (gr1) -- (gr0);\n  \\draw[->] (gr0) -- (gm);\n  \\draw[->] (pr1) -- node[left] {$\\alpha_{R^{\\oplus m}}$} (gr1);\n  \\draw[->] (pr0) -- node[left] {$\\alpha_{R^{\\oplus n}}$} (gr0);\n  \\draw[->] (pm) -- node[left] {$\\alpha_M$} (gm);\n \\end{tikzpicture}\n\\end{center}\nwhere $\\Gamma_g(L)$ denotes the graph of $g$ in $L \\otimes_k L$. The left and middle maps are isomorphisms, so the right map must also be an isomorphism. \n\\end{proof}\n\n\\begin{prop}[Descent for orbits]\\label{prop:objblockdescent}\nLet $X$ be a $k$-scheme, $L\/k$ a finite $G$-Galois extension,\nand $\\pi: X_L \\to X$ the natural projection map. If $\\mathsf{E} = \\{E_1,\n\\ldots, E_s\\}$ is a $G$-orbit forming an exceptional collection on $X_L$,\nand if $E$ is any element of $\\mathsf{E}$, then there is an exceptional\nobject $F$ in $\\mathsf{D^b}(X)$ such that $\\pi_*E \\simeq F^{\\oplus m}$\nand $\\pi^*F$ generates the category $ \\langle \\mathsf{E} \\rangle$.\n\\end{prop}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:collectiontoblocks}, exceptional $G$-orbits are\ncompletely orthogonal (and by definition carry a transitive action of\n$G$), which will be used throughout the proof. Fix an element $E \\in\n\\mathsf{E}$, so that $E = E_i$ for some $i$. Lemma~\\ref{lem:galconj}\ngives $$\\pi^*\\pi_*E \\simeq \\bigoplus _{g \\in G} g^*E$$  We claim that\n$\\text{End}(\\pi_*E)$ is a matrix algebra over a division algebra, and\nprove this by first showing that it is semisimple. Indeed, using\n$\\text{End}_X(M) \\otimes _k L \\simeq \\text{End}_{X_L} (\\pi^*M)$ for any $M\n\\in \\mathsf{D^b}(X)$ \\cite[Rem. 2.1]{AB}, we have\n\\[ \\text{End}_X(\\pi_*E) \\otimes _k L \\simeq\n\\text{End}_{X_L}(\\pi^* \\pi_* E) \\simeq\n\\text{End}_{X_L}\\left(\\bigoplus _{ g\\in G} g^*E\\right).\n\\]  Each $g^*E$ is exceptional so that\n$\\text{End}_{X_L}(g^*E) =: D_g$ is a division algebra for each element $g\n\\in G$.  Let  $H \\leq G$ be the subgroup consisting of elements $h$\nsatisfying $h^*E \\simeq E$. For any system of coset representatives $g\n\\in G\/H$, we have $\\text{End}_X(\\pi_*E)_L \\simeq \\prod _{g \\in\nG\/H}M_m(D_g)$, where $m = |H|$. This product of matrix algebras over\ndivision algebras is semisimple, i.e., the Jacobson radical\n$\\text{rad}(\\text{End}_X(\\pi_*E)_L) = 0$.  We then have $0 =\n\\text{rad}(\\text{End}_X(\\pi_*E)_L) = \\text{rad}(\\text{End}_X(\\pi_*E))_L$\nby ~\\cite[Thm.~1]{Amitsur}, and hence $\\text{rad}(\\text{End}_X(\\pi_*E))\n= 0$. Thus, $\\text{End}_X(\\pi_*E)$ is semisimple and so must also\nbe a product of matrix algebras over division algebras by the\nArtin-Wedderburn Theorem.  \n\nLet $Z$ be the center of $\\text{End}_X(\\pi_*E)$ and $Z_L$ the center of $\\text{End}_X(\\pi_*E)_L$.\nNote that $Z$ is an \\'{e}tale $k$-algebra, and to show that $\\text{End}(\\pi_*E)$ is a matrix algebra, it suffices to show that $Z$ has no zero divisors, and is thus a field.  There is an embedding $Z \\hookrightarrow Z_L  = \\prod_{g \\in G\/H}  L_g$, where $L_g$ is the center of the division algebra $D_g$.  The transitive action of $G$ on $\\{E_1,..., E_s\\}$ implies that $G$ acts transitively on a basis of $Z_L$, so that $Z = (Z_L)^G$ has no zero divisors.\n\nWe produce the object $F$ using the identification $\\text{End}_X(\\pi_*E)\n\\simeq M_n(D)$, where $D$ is a division algebra.\nLet $e_i = e_{ii}$ denote the usual idempotent matrices, so that\n$\\{e_i\\}$ is a complete set of primitive orthogonal idempotents.\nNotice that $F_i:= \\operatorname{Im} (e_i)$ is a simple $\\text{End}_X(\\pi_*E)$-submodule of $\\pi_*E$ for each $i$, and hence $F_i \\simeq F_j$ for each $i, j$, and $\\text{End}_X(F_i) \\simeq D$. Define $F = \\operatorname{Im} (e_1) \\subset \\pi_*E$, included as a direct summand.  We note that $\\pi_*E \\simeq \\bigoplus F_i \\simeq F^{\\oplus n}$.  \n\nWe now show that $F$ is an exceptional object on $X$.  As stated above, $\\text{End}_X(F)$ is a division algebra, so it suffices to show that $\\text{Ext}^n_X(F, F) = 0$ for $ n \\neq 0$.  Using Lemma~\\ref{lem:galconj} and $(\\pi^*, \\pi_*)$-adjunction, we have $$\\text{Ext}^n_X(\\pi_*E, \\pi_*E) = \\bigoplus _{g\\in G} \\text{Ext}^n _{X_L}(g^*E, E).$$  For $n \\neq 0$, each summand of the right-hand side is 0, which follows from the mutual orthogonality of the exceptional block $\\mathsf{E}$ (when $g^*E \\not\\simeq E$) and from exceptionality of $E$ (when $g^*E \\simeq E$).  Since $F$ is a direct summand of $\\pi_*E$, it follows that $\\text{Ext}^n_X(F, F)$ is a summand of $\\text{Ext}^n_X(\\pi_*E, \\pi_*E) = 0$.\n\nLastly, we show that $\\pi^*F$ generates the category $\\langle \\mathsf{E}\n\\rangle$.   Since $ F^{\\oplus m} \\simeq \\pi_*E$, extending scalars to $L$\ngives $ (\\pi^*F)^{\\oplus m} = \\pi^*(F^{\\oplus m}) \\simeq \\pi^*\\pi_*E\n\\simeq \\bigoplus g^*E$.  Thus, $$\\langle \\pi^*F \\rangle = \\langle (\\pi ^*F )^{\\oplus m} \\rangle = \\langle \\bigoplus g^* E \\rangle = \\langle g^*E \\rangle _{g \\in G} = \\langle \\mathsf{E} \\rangle.$$\\end{proof}\n\n\n\\begin{rem}\\label{rem:descSOD}\nProposition~\\ref{prop:objblockdescent} provides a very specific case of descent\nfor triangulated categories, the main advantage of which is that it\nallows one to identify a specific exceptional object that base extends\nto the given orbit.\nMoreover, a $G$-orbit which forms an exceptional collection consisting of vector bundles (resp. sheaves) descends to an exceptional collection consisting of vector bundles (resp. sheaves).  Compare to the following descent result for semiorthogonal decompositions, which generalizes \\cite[Cor. 2.15]{Toen}.  Although this result is useful for descending semiorthogonal decompositions, it does not identify exceptional objects.\n\n\\begin{prop}[Prop. 2.12, \\cite{AB}]\\label{prop:ABdesc}\nLet $\\mathsf{T}$ be a $k$-linear triangulated category such that $\\mathsf{T}_{k^s}$ is $k^s$-equivalent to $\\mathsf{D^b}(k^s, (k^s)^n)$.  Then there exists an \\'{e}tale algebra $K$ of degree $n$ over $k$, an Azumaya algebra $A$ over $K$, and a $k$-linear equivalence $\\mathsf{T} \\simeq \\mathsf{D^b}(K\/k, A)$.\n\\end{prop}\n\n\\noindent Let $X$, $\\mathsf{E}$, and $F$ be as in Proposition~\\ref{prop:objblockdescent}, and note that taking $\\mathsf{T} = \\langle F \\rangle$, we have $\\mathsf{T}_{k^s} = \\langle \\pi^* F \\rangle_{k^s} = \\langle \\mathsf{E} \\rangle _{k^s}$. Since $\\mathsf{E} = \\{ g^*E\\}_{g \\in G}$ is a full exceptional collection for $\\langle \\mathsf{E} \\rangle$, the bundle $\\mathcal{E} : = \\oplus (g^*E)_{k^s}$ is a tilting object for $\\langle \\mathsf{E}\\rangle_{k^s}$.  This defines an equivalence $$ \\mathsf{T}_{k^s} \\simeq \\langle \\mathsf{E} \\rangle_{k^s} \\simeq \\mathsf{D^b}(k^s, \\text{End}(\\mathcal{E})) = \\mathsf{D^b}(k^s, (k^s)^n).$$ Proposition~\\ref{prop:ABdesc} yields an \\'{e}tale extension $K\/k$, an Azumaya $K$-algebra $A$, and an equivalence $\\mathsf{T} \\simeq \\mathsf{D^b}(K\/k, A)$.  In this case, since $\\mathsf{T} = \\langle F \\rangle$, we see that $A = \\text{End}_X(F)$ is an Azumaya algebra over its center $Z$ (using the notation found in the proof of Proposition~\\ref{prop:objblockdescent}), which is simply a field extension of $k$.\n\\end{rem}\n\n\\begin{thm}[Descent for stable collections]\\label{thm:descblocks}\nLet $X$ be a $k$-scheme, $L\/k$ a finite $G$-Galois extension, and $\\pi: X_L \\to X$ the natural projection map.  If $X_L$ admits a full $G$-stable exceptional collection $\\mathsf{E}$ of objects of $\\mathsf{D^b}(X_L)$, then $X$ admits a full exceptional collection $\\mathsf{F}$ of objects of $\\mathsf{D^b}(X)$.  If $\\mathsf{E}$ is strong, so is $\\mathsf{F}$.  If the elements of $\\mathsf{E}$ are vector bundles (resp. sheaves), the elements of $\\mathsf{F}$ are vector bundles (resp. sheaves).\n\\end{thm}\n\n\\begin{proof}\n\nBy Lemma~\\ref{lem:collectiontoblocks}, we may write $\\mathsf{E} =\n\\{\\mathsf{E}^1,..., \\mathsf{E}^s\\}$ as a collection of $G$-stable\nblocks, where each block is given by a $G$-orbit.\nProposition~\\ref{prop:objblockdescent} then associates to each block\n$\\mathsf{E}^i$ an exceptional object $F_i$ on $X$, and we show that\n$\\mathsf{F} = \\{F_1,..., F_s\\}$ is a full exceptional collection on $X$.\nWe first show that $\\text{Ext}^n_{X}(F_i, F_j) = 0$ for all $n$ whenever\n$i > j$.  Let $E^i$ and $E^j$ be elements of the collections\n$\\mathsf{E}^i$ and $\\mathsf{E}^j$, respectively.  We then have\n\\begin{equation}\n\\text{Ext}^n_X(\\pi_*E^i, \\pi_*E^j) \\simeq\n\\bigoplus _{g\\in G} \\text{Ext}^n _{X_L}(g^*E^i, E^j).\n\\label{eq:1}\n\\end{equation}\nSince\n$E^i$ and $E^j$ are elements of the exceptional collection $\\mathsf{E}$\nand $i < j$, each summand is 0 for all $n$, so that\n$\\text{Ext}_X^n(\\pi_*E^i, \\pi_*E^j) = 0$ for all $n$.  The objects $F_i$\nand $F_j$ are direct summands of $\\pi_*E^i$ and $\\pi_*E^j$,\nrespectively, and it follows that $\\text{Ext}^n_X(F_i, F_j) = 0$ for all\n$n$.\n\nBy Proposition~\\ref{prop:exctosod}, the exceptional collection $\\{ F_1,\n\\ldots, F_s\\}$ yields a semiorthogonal decomposition\n\\[\n\\mathsf{D^b}(X) = \\langle \\mathscr{F}_1, \\ldots, \\mathscr{F}_s, \\mathsf{A}\\rangle,\n\\]\nwhere\n$\\mathscr{F}_i = \\langle F_i \\rangle$ and $\\mathsf{A}$ is the full\nsubcategory of objects $A$ with $\\text{Hom}_{\\mathsf{D^b}(X)}(A, F_i) =\n0$ for all $i$.  In particular, the subcategories $\\mathscr{F}_i$ are\nadmissible.  Extending scalars to $L$, we have $(\\mathscr{F}_i)_L =\n\\langle \\mathsf{E}^i\\rangle$, as both categories are generated by\n$\\pi^*F$ by Proposition~\\ref{prop:objblockdescent}. The exceptional collection\n$\\mathsf{E} =\\{\\mathsf{E}^1,\\ldots, \\mathsf{E}^s \\}$ is full, hence we have\na semiorthogonal decomposition $$ \\mathsf{D^b}(X_L) =  \\langle\n(\\mathscr{F}_1)_L, \\ldots, (\\mathscr{F}_s)_L \\rangle.$$  Since our\nadmissible subcategories $\\mathscr{F}_i$ base extend to a semiorthogonal\ndecomposition, \\cite[Lem. 2.9]{ABB} gives a semiorthogonal decomposition\n$\\mathsf{D^b}(X) = \\langle \\mathscr{F}_1, \\ldots, \\mathscr{F}_s \\rangle$.\nIn particular, the collection $\\{F_1, \\ldots, F_s\\}$ generates\n$\\mathsf{D^b}(X)$, so this collection is full.\n\nIf $\\mathsf{E}$ is strong, the right side of \\eqref{eq:1} vanishes for $i \\neq j$ (and any $n$).  It follows exactly as above that $\\text{Ext}^n_{X}(F_i, F_j) = 0$ for all $n$ when $i \\neq j$, so that $\\mathsf{F}$ is strong.\n\\end{proof}\n\n\\begin{rem}\\label{rem:elagin}\nSimilar descent results for collections of sheaves are given by Elagin in the algebraically closed case (i.e., $k = \\bar{k}$) using the framework of equivariant exceptional collections in equivariant derived categories \\cite{Elagin}.  Indeed, for a variety $X$ with an action of a finite group $G$ and a $G$-invariant exceptional collection (see Remark~\\ref{rem:invariant}) consisting of sheaves, this descent result is given in terms of $\\alpha$-twisted representations of $G$ (see Theorem 2.2 of loc. cit.).  For a $G$-stable exceptional collection consisting of sheaves, results are in terms of coinduced twisted representations of $G$ (see Theorem 2.3 of loc. cit.).\n\\end{rem}\n\n\\begin{lem} \\label{lem:Galconverse}\nLet $X$ be a $k$-scheme and $L\/k$ a finite $G$-Galois extension.  If $X$ admits an exceptional collection, then $X_L$ admits a $G$-stable exceptional collection.\n\\end{lem}\n\n\\begin{proof}\n Let $E_1,\\ldots,E_s$ be the exceptional collection on $X$ and consider $\\pi^\\ast E_1, \\ldots, \\pi^\\ast E_s$ on $X_L$. To compute morphisms, we note that \n \\begin{displaymath}\n  \\operatorname{Hom}_{X_L} (\\pi^\\ast E_i , \\pi^\\ast E_j ) = \\operatorname{Hom}_X( E_i, \\pi_\\ast \\pi^\\ast E_j) = \\operatorname{Hom}_X(E_i, E_j \\otimes_k L) = \\operatorname{Hom}_X(E_i,E_j) \\otimes_k L.\n \\end{displaymath}\n This vanishes if $j > i$. Let $A_i = \\operatorname{Hom}_X(E_i, E_i)$. We can split $A_i \\otimes_k L$ as a product of matrix algebras over division algebras $A_{i,j} = M_{N_{i,j}}(D_{i,j})$ and correspondingly decompose\n \\begin{displaymath}\n  \\pi^\\ast E_i = \\bigoplus F_{i,j}^{N_{i,j}}\n \\end{displaymath}\n with \n \\begin{displaymath}\n  \\operatorname{Hom}_{X_L}( F_{i,j} , F_{i,j} ) = D_{i,j}.\n \\end{displaymath}\n Note that $F_{i,j}$ and $F_{i,j^\\prime}$ are orthogonal for $j \\not = j^\\prime$. Thus, we have an exceptional collection.\n\\end{proof}\n\n\\begin{lem}\\label{lem:picinv}\nLet $X$ be a $k$-scheme and $L\/k$ a finite extension with Galois group $G$.  If $G$ acts trivially on $\\operatorname{Pic}(X_L)$ and $X_L$ admits an exceptional collection of line bundles, then $X$ admits an exceptional collection of vector bundles.\n\\end{lem}\n\n\\begin{proof}\n The collection on $X_L$ is automatically $G$-stable pointwise. Hence we can apply Theorem~\\ref{thm:descblocks}. \n\\end{proof}\n\n\\begin{rem}\n Note that while we may start with a collection of line bundles,\nthe descended collection may not consist only of line bundles.\nAn example of this is the real conic discussed in the introduction.\n\\end{rem}\n\n\\begin{lem}\\label{lem:blowup}\nLet $X$ be a smooth $k$-variety and $L\/k$ a $G$-Galois extension.  Let $Y_1, ..., Y_s$ be a $G$-orbit of smooth transversal subvarieties of $X_L$. Let $Y_I = \\cap_{i \\in I} Y_i$ and let $H_I$ be the normalizer of $Y_I$. If each $Y_I$ admits a full $H_I$-stable exceptional collection, then $\\tilde{X}$ admits an exceptional collection, where $\\tilde{X}_L$ is the iterated blow up of $X_L$ at the $Y_i$ (in any order).\n\\end{lem}\n\n\\begin{proof}\n This is an iterated application of Orlov's Theorem, see \\cite[Lemma 7.2]{CT}.\n\\end{proof}\n\n\n\n\\section{Arithmetic toric varieties}\\label{section:toric}\n\nWe introduce toric varieties over arbitrary fields. Such varieties, also known as \\emph{arithmetic toric varieties}, have been treated in \\cite{Duncan, ELFST, MerkPan, VosKly}.  \n\\begin{defn}\n\nA \\emph{torus} (over $k$) is an algebraic group $T$ (over $k$) such that $T _{k^s} \\simeq \\mathbb{G}_{m} ^n$. A torus is \\emph{split} if $T \\simeq \\mathbb{G}_{m} ^n$.  A field extension $L\/k$ satisfying $T_L \\simeq \\mathbb{G}_{m} ^n$ is called a \\emph{splitting field} of the torus $T$.  Any torus admits a finite Galois splitting field. \n\\end{defn}\n\n\\begin{defn}\nGiven a torus $T$, a \\emph{toric} $T$-\\emph{variety} is a normal variety with a faithful $T$-action and a dense open $T$-orbit.  A toric $T$-variety is \\emph{split} if $T$ is a split torus. A \\emph{splitting field} of a toric $T$-variety is a splitting field of $T$. A variety is a \\emph{toric variety} if it is a toric $T$-variety for some torus $T$.\n\\end{defn}\n\n\n\\begin{defn}\nLet $X$ be a toric $T$-variety whose dense open $T$-orbit contains a $k$-rational point.  Then we say $X$ is \\emph{neutral} \\cite{Duncan} (or a \\emph{toric} $T$-\\emph{model} \\cite{MerkPan}).\nAn orbit of a split torus always has a $k$-point, so a split toric\nvariety is neutral; but the converse is not true in general.\n\\end{defn}\n\n\\begin{rem}\nIn what follows, we will use the term \\emph{toric variety} to mean toric $T$-variety for some fixed torus $T$, even though such a variety may have a toric structure for various tori. In fact, the choice of torus does not affect our analysis of toric varieties given below, and we refer interested readers to \\cite{Duncan} for such considerations.\n\nRecall that a $k$-\\emph{form} of a $k$-variety $X$ is a $k$-variety $X'$ such that $X_L \\simeq X_L '$ for some field extension $L\/k$.  Any $k$-form of a toric variety is a toric variety \\cite{Duncan}.\n\\end{rem}\n\n\n\\subsection{The split case}  Let us begin by recalling some facts concerning toric varieties with $T \\simeq \\mathbb{G}_{m} ^n$ (e.g., when $k = {\\mathbb C}$ or $k= k^s$), which are studied in terms of combinatorial data, e.g., lattices, cones, fans.  Good references for toric varieties over ${\\mathbb C}$ include \\cite{Fulton, CLS}, and many results hold generally in the split case.\n\nLet $N$ be a finitely generated free abelian group of rank $n$ and $M = \\hom (N, {\\mathbb Z})$.  A subsemigroup $\\sigma \\subset N_{{\\mathbb R}}$ is a \\emph{cone} if ($\\sigma ^{\\vee})^{\\vee} = \\sigma$, where $\\sigma ^{\\vee} = \\{ u \\in M \\mid u(v) \\geq 0 \\text{ for all } v \\in \\sigma\\}$.  A subsemigroup $\\tau$ is a \\emph{face} of $\\sigma$ if it is of the form $\\tau = \\{v \\in \\sigma \\mid u(v) = 0 \\text{ for all } u \\in S \\}$ for some $S \\subseteq \\sigma ^{\\vee}$.  A cone $\\sigma$ is \\emph{pointed} if 0 is a face of $\\sigma$, and in this case $\\sigma^{\\vee}$ generates $M_{{\\mathbb R}}$.  Given a pointed cone $\\sigma$, we associate the affine $k$-scheme $U_{\\sigma} = \\operatorname{Spec} k[\\sigma ^{\\vee}]$, and for any face $\\tau \\subset \\sigma$ the induced map $U_{\\tau} \\hookrightarrow U_{\\sigma}$ is an open embedding.\n\nA \\emph{fan} $\\Sigma \\subset N_{{\\mathbb R}}$ is a finite collection of pointed cones such that (1) any face of a cone in $\\Sigma$ is a cone in $\\Sigma$ and (2) the intersection of any two cones in $\\Sigma$ is a face of each.  To any fan $\\Sigma$ we associate a $k$-variety $X_{\\Sigma}$ which is obtained by gluing the affine schemes $U_{\\sigma}$ along common subschemes $U _{\\tau}$ corresponding to faces.\n\nOn the other hand, beginning with a split torus $T \\simeq \\mathbb{G}_{m} ^n$ and toric $T$-variety $X$ with fixed embedding $T \\hookrightarrow X$, we recover $M$ as the character lattice $\\hom (T, \\mathbb{G}_{m})$ of $T$ and $N$ as the cocharacter lattice $\\hom (\\mathbb{G}_{m}, T)$.  The association $\\Sigma \\mapsto X_{\\Sigma}$ defines a bijective correspondence between fans $ \\Sigma \\subset N_{{\\mathbb R}}$ and toric $T$-varieties $X$ (we remind the reader that here we assume $T$ is a split torus;  in general, fans $\\Sigma$ admitting an action by $\\text{Gal}(k^s\/k)$ are in bijection with neutral toric $T$-varieties).\n\nLet $\\Sigma(\\ell)$ denote the collection of cones in $\\Sigma$ of dimension $\\ell$.  Let $\\text{Div}_T(X)$ denote the free abelian group generated by the \\emph{rays} of $\\Sigma$, i.e., elements of $\\Sigma (1)$.  By the Orbit-Cone Correspondence \\cite[Thm. 3.2.6]{CLS}, $\\text{Div}_T(X)$ is isomorphic to the group of $T$-invariant Weil divisors of $X$.  For $X$ a (split) smooth projective toric variety, we have natural identifications $\\text{Pic}(X) = \\text{Pic} (X_{k^s}) = \\text{Cl}(X_{k^s}) = \\text{Cl}(X)$ which yield an exact sequence $$0 \\to M \\to \\text{Div}_T(X) \\to \\text{Pic}(X) \\to 0.$$  In particular, if $X$ is of dimension $n$ and $m$ is the number of rays in $\\Sigma$, the Picard rank of $X$ is $\\rho = m-n$.\n\n\\begin{defn}\nA variety $X$ is \\emph{Fano} (resp. \\emph{weak Fano}) if its anticanoncial class $-K_X$ is ample (resp. nef and big).  If $X$ is a normal variety, a Cartier $D$ divisor on $X$ is \\emph{nef} (``numerically effective\" or ``numerically eventually free\") if $D \\cdot C \\geq 0$ for every irreducible curve $C \\subset X$.  A divisor $D$ is \\emph{very ample} if $D$ is base point free and $\\varphi_D : X \\to \\mathbb{P}(\\Gamma(X, \\O_X(D))^{\\vee})$ is an embedding. A divisor $D$ is \\emph{ample} if $\\ell D$ is very ample for some $\\ell \\in {\\mathbb Z}^+$.  A line bundle $\\O_X(D)$ is nef or (very) ample if the corresponding divisor $D$ is nef or (very) ample.  A Cartier divisor is \\emph{numerically trivial} if $D\\cdot C =0$ for every irreducible complete curve $C \\subset X$.  Let $N^1(X)$ be the quotient group of Cartier divisors by the subgroup of numerically trivial divisors.  The \\emph{nef cone} $\\text{Nef}(X)$ is the cone in $N^1(X)$ generated by the nef divisors, and the \\emph{anti-nef cone} is the cone $-\\text{Nef}(X) \\subset N^1(X)$. A line bundle $\\O_X(D)$ is nef (ample) if $D$ is nef (ample).  \n\\end{defn}\n\n\\begin{prop}\\label{prop:nef}\nA Cartier divisor $D$ on a split proper toric variety $X$ is nef (resp. ample) if and only if $D\\cdot C \\geq 0$ (resp. $D\\cdot C > 0$) for all torus-invariant integral curves $C \\subset X$.  \n\\end{prop}\n\n\\begin{proof}\nWhen $k$ is algebraically closed, these are Theorems 3.1 and 3.2 of \\cite{Mustata}. One can see that the arguments remain valid in the split case more generally.\n\\end{proof}\n\n\n\\subsection{The not necessarily split case}\n\nHere we provide a ``black box''\nfor producing exceptional collections on arbitrary forms of toric\nvarieties by identifying certain special exceptional collections on\na \\emph{split} toric variety.\nThis reduces an arithmetic question to a completely geometric question.\n\nWe begin by reviewing how to obtain arbitrary forms of toric varieties\nfrom the split case\n(see, for example, \\cite{Vos82Projective,ELFST}).\nLet $T$ be the split torus of a split smooth projective toric variety\n$X$ with fan $\\Sigma$ in the space $N \\otimes {\\mathbb R}$ associated to the\nlattice $N$.\nLet $\\operatorname{Aut}(\\Sigma)$ denote the subgroup of elements $g \\in \\operatorname{GL}(N)$\nsuch that $g(\\sigma) \\in \\Sigma$ for every cone $\\sigma \\in \\Sigma$.\nThere is a natural inclusion of $T \\rtimes \\operatorname{Aut}(\\Sigma)$\ninto $\\operatorname{Aut}(X)$ as the subgroup leaving the open orbit $T$-invariant.\n\nLet $k^s$ be the separable closure of $k$.\nThe Galois cohomology set $H^1(k^s\/k,\\operatorname{Aut}(X)(k^s))$ is in bijective\ncorrespondence with the $k$-forms of $X$.\nThe natural map\n\\[ H^1(k^s\/k, T(k^s) \\rtimes \\operatorname{Aut}(\\Sigma)) \\to H^1(k^s\/k, \\operatorname{Aut}(X)(L)) \\]\nin Galois cohomology is surjective;\nthe failure of this map to be a bijection amounts to the fact that there\nmay be several non-isomorphic toric variety structures\non the same variety (see \\cite{Duncan} for more details).\n\nSuppose $X'={}^\\gamma X$ is a twisted form of a split toric variety\nfor a cocycle $\\gamma$ representing a class in\n$H^1(k^s\/k, T(k^s) \\rtimes \\operatorname{Aut}(\\Sigma))$.\nThere is a ``factorization'' $X'={}^\\alpha({}^\\beta X)$\nwhere $\\beta$ represents a class in $H^1(k^s\/k, \\operatorname{Aut}(\\Sigma)$\nand $\\alpha$ represents a class in $H^1(k^s\/k, ({}^\\beta T)(k^s) )$.\nInformally, $\\beta$ changes the torus that acts on $X$,\nwhile $\\alpha$ changes the torsor of the open orbit in $X$.\n\nSuppose $X$ is a toric $T$-variety.\nWe say that an object $E \\in \\mathsf{D^b}(X)$ is \\emph{$T$-equivariant}\nif $E$ is in the image of the forgetful functor from\n$\\mathsf{D^b}(\\operatorname{Coh}_T(X))$\n(see \\S{2}~of~\\cite{BFK2}).\nIn particular, this implies that $t^\\ast E \\simeq E$ for all $t \\in\nT(k)$.\n\n\\begin{prop} \\label{prop:blackbox}\nLet $X$ be a split toric $T$-variety over a field $k$ and let $\\Sigma$\nbe the associated fan.  Suppose that $X$ admits an\n$\\operatorname{Aut}(\\Sigma)$-stable full exceptional collection $\\mathsf{E}$\nsuch that each object is $T$-equivariant.\nThen any $k$-form $X'$ of $X$ admits a full exceptional collection\n$\\mathsf{E'}$.\nMoreover, $\\mathsf{E'}$ is strong (resp. consists of vector bundles,\nconsists of sheaves) as soon as $\\mathsf{E}$ is strong\n(resp. consists of vector bundles, consists of sheaves).\n\\end{prop}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:Galconverse}, there exists a $G$-stable\nexceptional collection $\\mathsf{F}$ on $X_L$.\nFrom the proof of that lemma, the objects $F$ of $\\mathsf{F}$ are direct\nsummands of $\\pi^\\ast E$ for each object $E \\in \\mathsf{E}$,\nwhere each isomorphism class of simple direct summand is represented\nby exactly one $F$.\nSince $\\mathsf{E}$ is $\\operatorname{Aut}(\\Sigma)$-stable and each object\nis $T$-equivariant, we may conclude that $\\mathsf{F}$ is\n$(T(L) \\rtimes \\operatorname{Aut}(\\Sigma)) \\rtimes G$-stable.\n\nLet $X'$ be a $k$-form of $X$; there exists a finite Galois\nextension $L\/k$ with Galois group $G$ such that $X'_L \\simeq X_L$.\nFrom Theorem~5.1~of~\\cite{Duncan}, the natural map\n\\[ H^1(L\/k, T(L) \\rtimes \\operatorname{Aut}(\\Sigma)) \\to H^1(L\/k, \\operatorname{Aut}(X)(L)) \\]\nin Galois cohomology is surjective.\nThus, we may assume that $X' ={}^cX$ is the \\emph{twist}\nby a cocycle $c: G \\to T(L) \\rtimes \\operatorname{Aut}(\\Sigma)$.\nRecall that the cocycle condition is that\n$c(gh)=c(g){}^gc(h)$ for all $g,h \\in G$\nwhere ${}^gc(h)$ denotes the Galois action of $g$ on $T(L) \\rtimes \\operatorname{Aut}(\\Sigma)$.\n\nIdentifying $X_L=X'_L$, twisting gives $\\sigma'(g) = c(g) \\sigma(g)$\nwhere $\\sigma$ is the action of $G$ induced from $X$ and\n$\\sigma'$ is induced from $X'$.\nThe punchline is that the action $\\sigma'$ factors through the image of\n$(T(L) \\rtimes \\operatorname{Aut}(\\Sigma)) \\rtimes G$ described above.\nThus the exceptional collection $\\mathsf{F}$ is $G$-stable for the $X'$\naction as well.\nThe proposition now follows by Theorem~\\ref{thm:descblocks}.\n\\end{proof}\n\n\\begin{cor} \\label{cor:toricLB}\nLet $X$ be a split toric $T$-variety over a field $k$ and let $\\Sigma$\nbe the associated fan.\nIf $X$ admits an $\\operatorname{Aut}(\\Sigma)$-stable full (strong) exceptional collection of\nline bundles, then every $k$-form of $X$ admits a full (strong) exceptional\ncollection of vector bundles.\n\\end{cor}\n\n\\begin{proof}\nRecall that every line bundle is isomorphic to a $T$-equivariant line\nbundle by standard results on toric varieties.\nThe corollary now follows by Proposition~\\ref{prop:blackbox}.\n\\end{proof}\n\n\\begin{lem}\\label{lem:flips}\n Let $X$ and $Y$ be smooth projective toric varieties over $k$. Let $G = \\operatorname{Gal}(k^s\/k)$. Assume we have a $K$-positive toric flip $X \\dashrightarrow Y$ such that over $k^{s}$ the flipping loci $F_i$ are disjoint and permuted by $G$. Let $H_i$ be the normalizer of $F_i$. If $X_L$ admits a full $G$-stable exceptional collection and $Y_i$ admits a full $H_i$-stable exceptional collection, then $Y$ admits a full exceptional collection.\n\\end{lem}\n\n\\begin{proof}\n Passing to $k^{s}$ we are free to use \\cite{BFK} giving semi-orthogonal decompositions for the flip over each $Y_i$. Since the $Y_i$ are disjoint, we can concatenate these collections to get a $G$-stable collection. \n\\end{proof}\n\n\\subsection{Products of toric varieties}\n\nRecall that, given groups $G,H$ along with a homomorphism\n$\\rho : H \\hookrightarrow S_n$,\nthe \\emph{wreath product} $G \\wr H$ is the group $G^n \\rtimes H$\nwhere $H$ acts on $G^n$ by permuting the copies of $G$.\nWe say a toric variety $X$ is \\emph{indecomposable} if it cannot be written as\na product $X_1 \\times X_2$ where $X_1$ and $X_2$ are\npositive-dimensional toric varieties.\n\n\\begin{lem}\\label{lem:autprod}\nSuppose $Z= X_1^{n_1} \\times \\cdots \\times X_r^{n_r}$ is a product of\nproper split toric varieties $X_1, \\ldots, X_r$, where $X_i \\not\\simeq X_j$\nfor $i \\ne j$ and each $X_i$ is indecomposable.\nThen\n\\[ \\operatorname{Aut}(\\Sigma) \\simeq (\\operatorname{Aut}({\\Sigma_1}) \\wr S_{n_1}) \\times \\cdots \\times\n(\\operatorname{Aut}({\\Sigma_r}) \\wr S_{n_r}) , \\]\nwhere $\\Sigma$ is the fan of $Z$ and $\\Sigma_1, \\ldots, \\Sigma_r$ are\nthe fans of $X_1, \\ldots, X_r$.\n\\end{lem}\n\n\\begin{proof}\nFirst, consider $Z=X_1 \\times X_2$ where\n$X_1,X_2$ are proper split toric varieties.\nLet $N$ (resp. $N_1, N_2$) be the cocharacter lattice and\n$\\Sigma$ (resp. $\\Sigma_1, \\Sigma_2$) be the fan of\n$Z$ (resp. $X_1, X_2$).\nHere $N = N_1 \\oplus N_2$ and $\\Sigma$ is the set of cones\nof the form $\\sigma_1 \\times \\sigma_2$ where $\\sigma_1 \\in \\Sigma_1$\nand $\\sigma_2 \\in \\Sigma_2$.\nThe faces of a cone $\\sigma_1 \\times \\sigma_2$ are precisely\nthe cones of the form $\\sigma_1' \\times \\sigma_2'$ where\n$\\sigma_1'$ is a face of $\\sigma_1$ and $\\sigma_2'$ is a face of\n$\\sigma_2$.\nThe fan $\\Sigma_1$ can be canonically identified with the subfan of\n$\\Sigma$ via the bijection $\\sigma \\mapsto \\sigma \\times \\{0\\}$.\n\nNow, suppose also that $Z= Y \\times W$ is a product of proper split\ntoric varieties where $Y$ is indecomposable.  Let $\\Sigma_Y$ be the fan\nof $Y$, which we can canonically identify with a subfan of $\\Sigma_Z$.\nEvery cone of $Y$ is of the form $\\sigma_1 \\times \\sigma_2$ where\n$\\sigma_1 \\in \\Sigma_1$ and $\\sigma_2 \\in \\Sigma_2$.\nSince fans are closed under taking faces, $\\sigma_1 \\times \\{0\\}$\nand $\\{0\\} \\times \\sigma_2$ are also cones in $\\Sigma_Y$.\nThus every cone in $\\Sigma_Y$ is a product of cones in the intersections\n$\\Sigma_Y \\cap \\Sigma_1$ and $\\Sigma_Y \\cap \\Sigma_2$.\n\nIn particular, since $X$ is proper, we have that the space\n$N_Y \\otimes {\\mathbb R}$ is the direct sum of\n$(N_Y \\otimes {\\mathbb R}) \\cap (N_1 \\otimes {\\mathbb R})$ and\n$(N_Y \\otimes {\\mathbb R}) \\cap (N_2 \\otimes {\\mathbb R})$,\nand $\\Sigma_Y$ is a product of the fans $\\Sigma_Y \\cap \\Sigma_1$\nand $\\Sigma_Y \\cap \\Sigma_2$.\nSince $Y$ is indecomposable, one of these fans is indecomposable\nand $\\Sigma_Y$ must be a subfan of either $\\Sigma_1$ or $\\Sigma_2$.\n\nReturning to the general case, we conclude that the decomposition\n$\\Sigma= \\Sigma_1^{n_1} \\times \\cdots \\times \\Sigma_r^{n_r}$\nis unique up to ordering.\nThe description of the automorphism group is immediate.\n\\end{proof}\n\n\\begin{lem}\\label{lem:stabprod}\nLet $Z$ be a proper toric $k$-variety with splitting field $L\/k$.\nSuppose $Z_L = \\prod_{i=1}^n X_i$ where each $X_i$ is an indecomposable\nsplit proper toric $L$-variety admitting a full (strong)\n$\\operatorname{Aut}(\\Sigma_i)$-stable exceptional collection of line bundles,\nwhere $\\Sigma_i$ is the fan of $X_i$.\nThen $Z$ has a full (strong) exceptional collection of vector bundles.\n\\end{lem}\n\n\\begin{proof}\nIt is a well known that the exterior product collection is an\nexceptional collection.\nFor each isomorphism class among the $X_i$ fix a full (strong)\n$\\operatorname{Aut}(\\Sigma_{X_i})$-stable exceptional collection of line bundles.\nThis ensures that the exterior product collection is\nstable under the action of\n$(\\operatorname{Aut}(\\Sigma_{X_1}) \\wr S_{a_1}) \\times \\cdots \\times (\\operatorname{Aut}(\\Sigma_{X_r})\n\\wr S_{a_r})$.\nSince this group is $\\operatorname{Aut}(\\Sigma)$ by Lemma~\\ref{lem:autprod},\nthe exterior product collection descends by Corollary~\\ref{cor:toricLB}.\n\\end{proof}\n\n\n\\section{Low dimension or high symmetry}\\label{section:minimal}\n\nWe provide exceptional collections for smooth toric surfaces, Fano 3-folds,\nsome Fano 4-folds, centrally-symmetric toric varieties, and\ntoric varieties corresponding to root systems of type $A$. \n\n\n\\subsection{Surfaces}\\label{sect:surfaces}\nHere we prove that every toric surface has a full exceptional\ncollection.\nWe begin by recalling the (classical) minimal model program for surfaces\nover non-closed fields.\n\nSuppose $f : X \\to X'$ is a birational morphism of smooth projective\nsurfaces over a field $k$.\nIf $k$ is separably closed, then by Proposition~5~of\\cite{Coombes}\nthe morphism factors into a sequence\n\\[\nX= X_0 \\to X_1 \\to \\cdots \\to X_r = X'\n\\]\nwhere each morphism $X_i \\to X_{i+1}$ is the blowup of a point on\n$X_{i+1}$.\nOver a non-closed field $k$, we can factor $f : X \\to X'$ into a\nsequence where each morphism $X_i \\to X_{i+1}$ is defined over $k$\nand is a blowup of a\n(necessarily finite) Galois orbit of $k^s$-points on $X_{i+1}$.\n\nBlowing up a point produces an exceptional curve: a smooth rational\ncurve with self-intersection $-1$.  By Castelnuovo's contractibility\ncriterion, such a curve can always be obtained as the result of a blow-up.\nIf one finds a skew Galois orbit of such curves on $X$, then there\nexists a birational morphism $f : X \\to X'$ contracting these curves.\nRepetition of this procedure eventually terminates.\n\n\\begin{defn}\nA \\emph{minimal surface} $X$ is a smooth projective surface over a field $k$\nsuch that every birational morphism $X \\to X'$ to a smooth projective\nsurface $X'$ is an isomorphism.\n\\end{defn}\n\nAny smooth projective surface can be obtained by iteratively blowing up\nGalois orbits of separable points starting from a minimal model.\nA toric variety is geometrically rational.\nMinimal geometrically rational surfaces were classified by\nManin~\\cite{Manin} and Iskovskikh~\\cite{Iskovskikh}.\nOne checks that the toric surfaces in their collection are the\nfollowing (see also a direct proof in~\\cite{Xie}):\n\n\\begin{lem}\\label{lem:classification}\nA minimal smooth projective toric surface\nis a $k^s\/k$-form of one of the following:\n\\begin{enumerate}\n\\item $\\mathbb{P}^2$, $\\operatorname{Aut}(\\Sigma) = S_3$.\n\\item ${\\mathbb P}^1 \\times {\\mathbb P}^1$, $\\operatorname{Aut}(\\Sigma) = D_8$.\n\\item $\\mathbb{F}_{a} = \\operatorname{Proj}(\\O_{{\\mathbb P}^1} \\oplus \\O_{{\\mathbb P}^1} (a))$,\n$a \\geq 2$, $\\operatorname{Aut}(\\Sigma) = C_2$.\n\\item $\\mathsf{dP}_6 = $ del Pezzo surface of degree 6, $\\operatorname{Aut}(\\Sigma) = D_{12}$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nA minimal geometrically rational surface is either a del Pezzo surface\nor has a conic bundle structure \\cite{Manin,Iskovskikh}.\nOver the separable closure, a del Pezzo surface is either\n${\\mathbb P}^1 \\times {\\mathbb P}^1$ or a blow up of ${\\mathbb P}^2$ at up to $8$ points in\ngeneral position.\nBlowing up only one or two points never results in a minimal surface, and\nno more than three points can be simultaneously torus invariant and\nin general position.  Thus every del Pezzo surface is a $k^s\/k$-form\nof ${\\mathbb P}^2$, ${\\mathbb P}^1 \\times {\\mathbb P}^1$ or $\\mathsf{dP}_6$.\nOver the separable closure, a conic bundle structure has at most $2$\nsingular fibers since their images must be torus invariant points on the\nbase ${\\mathbb P}^1$.\nA minimal conic bundle with at most two singular fibers over the\nseparable closure must be either a del Pezzo surface or a minimal ruled\nsurface.\n\\end{proof}\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (-5,-5) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{$\\mathbb{P}^2$}};\n \\end{tikzpicture}\n\\hspace{1cm}\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (0,0) -- (0,-5);\n   \\draw (-5,0) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{${\\mathbb P}^1 \\times {\\mathbb P}^1$}};\n \\end{tikzpicture}\n\n\\vspace{0.5cm}\n\n\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (0,0) -- (0,-5);\n   \\draw (-2.5,5) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{$\\mathbb{F}_a$}};\n \\end{tikzpicture}\n\\hspace{1cm}\n \\begin{tikzpicture}\n   [scale=.4, vertex\/.style={circle,draw=black!100,fill=black!100, inner sep=0.5pt,minimum size=0.5mm}]\n   \\filldraw[fill=black!20!white,draw=white!100]\n     (-5,5) -- (5,5) -- (5,-5) -- (-5,-5) -- (-5,5);\n   \\draw (0,0) -- (5,0);\n   \\draw (0,0) -- (0,5);\n   \\draw (0,0) -- (0,-5);\n   \\draw (0,0) -- (-5,0);\n   \\draw (5,5) -- (0,0);\n   \\draw (-5,-5) -- (0,0);\n  \\foreach \\x in {-5,-4,...,5}\n   \\foreach \\y in {-5,-4,...,5}\n   {\n     \\node[vertex] at (\\x,\\y) {};\n   }\n\\node at (0,-6,0) {\\text{$\\mathsf{dP}_6$ }};\n \\end{tikzpicture}\n\\vspace{-.3cm}\n\\caption{Fans for minimal toric surfaces}\\label{fig:fans}\n\\end{center}\n\\end{figure}\n\nHere we exhibit full strong exceptional collections consisting of\n$G$-stable blocks for each minimal toric surface exhibited above\n(none of these collections are original).\nThe fans associated to the split forms of these surfaces are given in Figure~\\ref{fig:fans}. In each case, we fix a torus $T$ which gives $X$ the structure of a toric $T$-surface.  As remarked above, this gives a homomorphism $G \\to \\text{Aut}(\\Sigma)$ as well as an action of $G$ on $\\text{Pic}(X_L)$, where $L$ is a splitting field of $T$, $G = \\text{Gal}(L\/k)$, and $\\Sigma$ is the fan corresponding to the split toric surface $X_L$. We produce $G$-stable exceptional collections in each case by exhibiting $\\text{Aut}(\\Sigma)$-stable collections.\n\n\\begin{ex}\\label{ex:p2}\nLet $X$ be a toric $T$-surface whose split form is $\\mathbb{P}^2$ with\n$\\text{Aut}(\\Sigma) = S_3$. The $ S_3$-action on\n$\\text{Pic}(\\mathbb{P}^2) = {\\mathbb Z}$ is clearly trivial, so that the\nexceptional collection $\\{ \\O, \\O(1), \\O(2)\\}$, given in\n\\cite{Beilinson} yields a full strong $\\text{Aut}(\\Sigma)$-stable\nexceptional collection.  By Corollary~\\ref{cor:toricLB},\n$X$ admits a full strong exceptional collection.\n\\end{ex}\n\n\\begin{ex}\\label{ex:p1p1}\nLet $X$ be a toric surface whose split form is ${\\mathbb P}^1 \\times {\\mathbb P}^1$ with $\\text{Aut}(\\Sigma) = D_8$, and consider the natural projections $p_1, p_2: {\\mathbb P}^1 \\times {\\mathbb P}^1 \\to {\\mathbb P}^1$.  Let $\\O(p, q) = p_1 ^*\\O (p) \\otimes p_2^* \\O (q)$.  By \\cite{KvichanskyNogin}, the collection $\\{\\O, \\O(1, 0), \\O(0, 1), \\O(1,1) \\}$ on ${\\mathbb P}^1 \\times {\\mathbb P}^1$ is exceptional since $\\{ \\O, \\O(1)\\} $ is an exceptional collection for ${\\mathbb P}^1$. The $ D_8$-action preserves this collection, with orbits given by the blocks $\\mathsf{E}^0 = \\{ \\O \\}$, $\\mathsf{E}^1 = \\{ \\O(1, 0), \\O(0,1) \\} $, and $\\mathsf{E}^2= \\{ \\O(1,1)\\}$.  In particular, this collection above is $\\text{Aut}(\\Sigma)$-stable, and Corollary~\\ref{cor:toricLB} yields an exceptional collection on $X$.\n\\end{ex}\n\n\\begin{ex}\nLet $X$ be a toric surface whose split form is the Hirzebruch surface\n$\\mathbb{F}_a$; here $\\text{Aut}(\\Sigma) = C_2$.\nLet $e_1, e_2$ be the standard basis for ${\\mathbb Z}^2$.\nAs in \\cite[Ex. 4.1.8]{CLS},\nlet $u_1  =-e_1 + ae_2$, $u_2 = e_2$, $u_3 = e_1$, and $u_4\n= -e_2$ be the generators of $\\Sigma(1)$ with corresponding toric\ndivisors $D_i$.  The Picard group of $\\mathbb{F}_a$ is freely generated\nby $\\{D_1, D_2\\}$ and $D_1$ is linearly equivalent to $D_3$.\nThe only nontrivial fan automorphism $\\sigma$ takes $e_1 \\mapsto -e_1+ae_2$\nand $e_2 \\mapsto e_2$.\nThus $\\sigma$ leaves $D_2,D_4$ fixed and interchanges $D_1$ and $D_3$.\nWe conclude the action of $C_2$ on\n$\\text{Pic}(\\mathbb{F}_a)$ is trivial, and thus, any exceptional\ncollection is necessarily $G$-stable (see Lemma~\\ref{lem:picinv}).\nAn exceptional collection for $\\mathbb{F}_a$ is given by $\\{\\O, \\O(D_3),\n\\O(D_4), \\O(D_3 + D_4)\\}$ \\cite{KvichanskyNogin}.\nCorollary~\\ref{cor:toricLB} then gives an exceptional collection on $X$.\n\\end{ex}\n\n\\begin{ex}\\label{ex:dp6}\nLet $X$ be a toric surface whose split form is $\\mathsf{dP}_6$; here\n$\\text{Aut}(\\Sigma) = D_{12}$. Viewing $\\mathsf{dP}_6$ as the blowup of\n$\\mathbb{P} ^2$ at 3 non-colinear points, let $H$ be the pullback of the\nhyperplane divisor on $\\mathbb{P}^2$ and $E_i$ the exceptional divisors, $i =\n1, 2, 3$.  As shown in \\cite[Prop. 6.2(ii)]{King}, the collection\n$$\\{\\O, \\O(H - E_1), \\O(H - E_2), \\O(H - E_3), \\O(H), \\O(2H - (E_1 + E_2\n+ E_3)) \\}$$ gives an exceptional collection for $\\mathsf{dP}_6$,\nwhich is $\\text{Aut}(\\Sigma)$-stable.\n\nLet us rephrase this in the notation of \\cite{BSS}.\nThere are two morphisms $\\mathsf{dP}_6 \\to \\mathbb{P}^2$ realizing\n$\\mathsf{dP}_2$ as a blowup of $\\mathbb{P}^2$, and we denote the collection of\nall six exceptional divisors by $L_i$ and $M_i$, with $i = 1, 2, 3$.\nLet $H$ and $H'$ denote the pullbacks of the hyperplane divisors on\n$\\mathbb{P}^2$ under the maps contracting $M_i$ and $L_i$, respectively, where\nwe identify $H$ with the divisor given in King's collection above (and\nthus we also identify $E_i$ with $M_i$).\nThen $H = L_1 + M_2 + M_3$, and it follows that $$2H - (E_1 + E_2 + E_3)\n= L_1 + L_2 + M_3 = H'$$ using the relation $L_i + M_j = L_j + M_i$.\nFurthermore, one checks that $ H - E_1 =  L_2 + M_3$, $H-E_2 = L_1 +\nM_3$, and $H- E_3  = L_1 + M_2$.\nAs described in \\cite[$\\S$2]{BSS}, the element $\\sigma$ in  $S_3 \\times\nC_2 = D_{12}$ which cyclically permutes the six lines $L_i, M_i$ also\nsatisfies $\\sigma (H) = H'$ and $\\sigma^2(H) = H$.\nWe arrange the exceptional collection above into blocks $\\mathsf{E}^0 =\n\\{\\O \\}$, $\\mathsf{E}^1 = \\{ \\O(H - E_1), \\O(H - E_2), \\O(H - E_3)\\}$\nand $\\mathsf{E}^2 = \\{\\O(H), \\O(2H - (E_1 + E_2 + E_3))\\}$.\nIn particular, the exceptional collection given above is\n$\\text{Aut}(\\Sigma)$-stable, and by Corollary~\\ref{cor:toricLB} we have an exceptional\ncollection on $X$.\n\\end{ex}\n\n\\begin{prop}\\label{prop:surface}\nEvery toric surface admits a full exceptional collection of sheaves.\n\\end{prop}\n\n\\begin{proof}\nThere is a sequence of blowups $X = X_0 \\to \\cdots \\to X_s = X'$ where\n$X'$ is minimal, so must be one of the varieties given in\nLemma~\\ref{lem:classification}.\nBy Examples~\\ref{ex:p2}-\\ref{ex:dp6},\n$X'$ admits a full strong exceptional collection of vector bundles, and\nthus $X'_L$ admits a $G$-stable exceptional collection.  By\nLemma~\\ref{lem:blowup}, $X_L$ admits a $G$-stable exceptional\ncollection.\n\\end{proof}\n\n\\begin{rem}\nThe authors would like to thank F.~Xie for pointing out a mistake\nin the statement of a previous version of\nProposition~\\ref{prop:surface}.\nXie also discusses exceptional collections of toric surfaces in\n\\cite{Xie}, although her definition of exceptional object is not the\nsame as ours. \nIn the second arXiv version of that paper, Xie sketched in Remark~8.8\nhow one might construct an exceptional collection for toric surfaces.\nAfter the authors posted a preliminary version of this paper to the\narXiv, Xie updated her preprint with Corollary~8.8, which proves the analog of\nthe above proposition for collections of vector bundles but using her notion of exceptional collection.\n\\end{rem}\n\n\n\\subsection{The toric Frobenius and toric Fano 3-folds}\\label{sect:3fold}\nIn Table~\\ref{tab:3-folds} we present the classification of smooth toric Fano 3-folds given in \\cite{Batyrev, Watanabe}, adopting Batyrev's enumeration. For each $X = X_{\\Sigma}$, we record the following invariants:\n\\begin{itemize}\n\n\\item $\\sigma(1) = | \\Sigma (1) |$ is the number of rays of $\\Sigma$ \\cite{BT}.\n\n\\item $k_0$ is the rank of the Grothendieck group $K_0(X)$, which coincides with the number of maximal cones in the fan $\\Sigma$ \\cite{BT}.\n\n\\item $\\operatorname{Aut}(\\Sigma) $ is the automorphism group of the (lattice $N$ which preserves the) fan $\\Sigma$ corresponding to $X$.\n\n\\item $\\rho$ is the Picard rank of $X$ \\cite{Watanabe}.\n\n\\item $\\rho ^G$ is the $\\operatorname{Aut}(\\Sigma)$-invariant Picard rank of $X$,\ni.e., the rank of $\\text{Pic}(X)^{\\operatorname{Aut}(\\Sigma)}$.\n\n\\item $\\mathfrak{fr} = | \\mathsf{Frob}(X) |$ is the number of\nisomorphism classes of line bundles produced by the push forward of the structure sheaf under the Frobenius morphism \\cite{BT, Uehara}.\n\n\\item $ \\mathfrak{fr}^- = |\\mathsf{Frob}(X) \\cap -\\text{Nef}(X)|$ is the\nnumber of isomorphism classes of line bundles in $\\mathsf{Frob}(X)$ which lie in the anti-nef cone of $X$ \\cite{Uehara}.\n\\end{itemize}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lrlccccccc}\n\\toprule\n  & &Toric Fano 3-fold $X$ & $\\sigma(1)$ &$k_0$ & $\\operatorname{Aut}(\\Sigma)$ & $\\rho$ & $\\rho^G$ & $\\mathfrak{fr}$ & $\\mathfrak{fr}^- $\\\\\n\\midrule\n\n& 1. & $\\mathbb{P}^3$ & 4 &  4 & $S_4$ & 1 & 1 & 4 & 4  \\\\\n\n&2.  &$\\mathbb{P}_{\\mathbb{P}^2}(\\O \\oplus \\O(2))$ & 5 & 6 & $S_3$ & 2 & 2 & 7 & 6 \\\\\n\n &3.  &$\\mathbb{P}_{\\mathbb{P}^2}(\\O \\oplus \\O(1))$ & 5 & 6 & $S_3$ & 2 & 2 & 6 & 6\\\\\n \n&4.& $\\mathbb{P}_{\\mathbb{P}^1}(\\O \\oplus \\O \\oplus \\O(1))$ & 5 & 6 & $C_2 \\times C_2$ & 2 & 2 & 6 & 6\\\\\n\n &5. & $\\mathbb{P}^2\\times \\mathbb{P}^1$ & 5 & 6 & $D_{12}$ & 2 & 2 & 6 & 6\\\\\n \n &6.  &$\\mathbb{P}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(\\O\\oplus \\O(1,1))$ & 6 & 8 & $D_8$ & 3 & 2 & 8 & 8\\\\\n \n &7.  &$\\mathbb{P}_{\\mathsf{dP}_8}(\\O\\oplus \\O(l))$, $l^2=1$ on $\\mathsf{dP}_8$ & 6 & 8 & $D_8$ & 3 & 3 & 8 & 8 \\\\\n\n&8.  &$\\mathbb{P}^1\\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ & 6 & 8 & $C_2 \\times S_4$ & 3 & 1 & 8 & 8\\\\\n\n &9.  &$\\mathsf{dP}_8\\times \\mathbb{P}^1 $ & 6 & 8 &$ C_2 \\times C_2$ & 3 & 3 & 8& 8 \\\\\n \n&10. &$\\mathbb{P}_{\\mathbb{P}^1\\times\\mathbb{P}^1}(\\O\\otimes \\O(1,-1))$& 6 & 8 & $D_8$ & 3 & 2 & 8 & 8\\\\\n\n &11.  & $\\text{Bl}_{\\mathbb{P}^1}(\\mathbb{P}_{\\mathbb{P}^2}(\\O \\oplus \\O(1)))$& 6 & 8 & $C_2$ & 3 & 3 & 9 & 8 \\\\\n \n &12. & $\\text{Bl}_{\\mathbb{P}^1}(\\mathbb{P}^2\\times \\mathbb{P}^1)$& 6 & 8 & $C_2$ & 3 & 3 & 8 & 8 \\\\\n\n& 13.  & $\\mathsf{dP}_7-$bundle over $\\mathbb{P}^1$ & 7 & 10 & $C_2$ & 4 & 4 & 10 & 10 \\\\\n\n & 14. & $\\mathsf{dP}_7-$bundle over $\\mathbb{P}^1$ & 7 & 10 & $C_2 \\times C_2$ & 4 & 3 & 10 & 10\\\\\n\n& 15. & $\\mathsf{dP}_7\\times \\mathbb{P}^1$& 7 & 10 & $C_2 \\times C_2$ & 4 & 3 & 10 & 10\\\\\n\n& 16. & $\\mathsf{dP}_7-$bundle over $\\mathbb{P}^1$& 7 & 10 & $C_2$ & 4 & 4 & 10 & 10\\\\\n\n & 17.  &$\\mathsf{dP}_6\\times \\mathbb{P}^1$& 8 & 12 & $C_2 \\times C_2 \\times S_3$ & 5 &  2 & 12 & 12\\\\\n\n & 18. & $\\mathsf{dP}_6-$bundle over $\\mathbb{P}^1$& 8 & 12 & $C_2 \\times C_2$  & 5 & 4 & 12 & 12\\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{.2cm}\n\\caption{Toric Fano 3-folds}\\label{tab:3-folds}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Toric Frobenius}\\label{section:frob}\n\nLet $X$ be a split toric variety of dimension $n$ with fixed torus embedding $T \\hookrightarrow X$ and take $\\ell \\in {\\mathbb Z}^+$.  Define the $\\ell^{\\text{th}}$ Frobenius map on $T = \\mathbb{G}_{m}^n$ to be $(x_1,..., x_n) \\mapsto (x_1^{\\ell},..., x_n^{\\ell})$.  The unique extension to $X$ will be denoted $F_{\\ell}$ and called the \\emph{$\\ell^{th}$ Frobenius morphism}.  Alternatively, if $\\Sigma \\subset N$ is the fan associated to $X$, define a lattice $N' = \\frac{1}{\\ell} N$.  The inclusion $N \\subset N'$, which sends a cone in $N_{{\\mathbb R}}$ to the cone with the same support in $N'_{{\\mathbb R}}$, induces a finite surjective morphism which is precisely the $\\ell^{\\text{th}}$ Frobenius morphism $F_{\\ell}: X \\to X.$ \n\nThe sheaf $(F_{\\ell})_*(\\O_X)$ splits into line bundles and Thomsen\nprovides an algorithm for computing its direct summands  \\cite{Thomsen}.\nWe let $\\mathsf{Frob}(X)$ denote the union of all isomorphism classes of line bundles arising as direct summands of $(F_{\\ell})_*(\\O_X)$ as $\\ell$ varies over ${\\mathbb Z}^+$.  Note that $\\mathsf{Frob}(X)$ is a finite set. \n\n\\begin{conj}[Bondal \\cite{Bondal2}]\\label{conj:Bondal}\nIf $X$ is a smooth proper toric variety then the collection $\\mathsf{Frob}(X)$ generates $\\mathsf{D^b}(X)$.\n\\end{conj}\n\nFor a toric variety $X$ in which Bondal's Conjecture is true, we will\nsay that \\emph{the Frobenius generates the derived category of} $X$.\nIn loc. cit., Bondal proves that if all summands of $\\mathsf{Frob}(X)$ are nef, one actually gets a full strong exceptional collection, so that Conjecture~\\ref{conj:Bondal} is true in this case. He also notes his arguments work for all but two (isomorphism classes of) toric Fano threefolds. To cover all toric Fano threefolds, \nUehara noticed that discarding line bundles which do\nnot lie in the set $-\\text{Nef}(X)$ yields a full strong\nexceptional collection \\cite{Uehara}.\n\n\\begin{lem}\\label{lem:FrobNef}\nLet $X$ be a toric variety over $k$ with splitting field $L$.\nSuppose $\\mathsf{E}$ is a full (strong) exceptional collection for\n$\\mathsf{D^b}(X_L)$\nwhere either $\\mathsf{E} = \\operatorname{\\mathsf{Frob}}(X_L)$ or\n$\\mathsf{E} = \\operatorname{\\mathsf{Frob}}(X_L) \\cap - \\operatorname{Nef}(X_L)$.\nThen there exists a full (strong) exceptional collection for\n$\\mathsf{D^b}(X)$.\n\\end{lem}\n\n\\begin{proof}\nBoth $\\operatorname{\\mathsf{Frob}}(X_L)$ and $\\text{Nef}(X_L)$ are\ncanonical constructions and thus are $\\text{Aut}(X_L)$-stable.\nIn particular, $\\mathsf{E}$ is $\\text{Aut}(\\Sigma)$-stable\nand so Corollary~\\ref{cor:toricLB} applies.\n\\end{proof}\n\n\\begin{prop}\\label{prop:3fold}\nLet $X$ be a smooth projective toric Fano 3-fold over a field $k$.  Then $X$ admits a full strong exceptional collection consisting of vector bundles.\n\\end{prop}\n\n\\begin{proof}\nLet $X_L$ be the associated split toric Fano 3-fold.  The main result of \\cite{Uehara} guarantees that the set $\\mathsf{E} = \\mathsf{Frob}(X_L) \\cap - \\text{Nef}(X_L)$ defines a full strong exceptional collection on $X$. Lemma~\\ref{lem:FrobNef} completes the proof.\n\\end{proof}\n\n\n\\subsection{Toric Fano 4-folds}\\label{sect:4fold}\n\nThere are 124 split smooth toric Fano 4-folds,\nwhich were first classified in \\cite{Batyrev}\n(a missing case was added in~\\cite{Sato}).\nIn \\cite{Prabhu}, Prabhu-Naik exhibits full strong exceptional\ncollections for all 124 of these 4-folds.\nHowever, it is not clear that these collections are\n$\\operatorname{Aut}(\\Sigma)$-stable, so they do not necessarily lead to full strong\nexceptional collections in the arithmetic case.\n\nAll collections obtained\nusing Method 1 of loc. cit. produce $\\operatorname{Aut}(\\Sigma)$-stable collections (note that\nthis is precisely the method used in \\cite{Uehara} for toric Fano\n3-folds, and we will refer to this as the \\emph{Bondal-Uehara Method}).\nTogether with Lemmas~\\ref{lem:stabprod} and \\ref{lem:FrobNef}, this gives stable exceptional collections for 43 of the 124 smooth toric Fano 4-folds.  However, there are examples when the\nBondal-Uehara Method fails to produce an exceptional collection.\nIn this case, all is not lost (see Section~\\ref{sec:centsym}).\n\nMore precisely, the varieties (61), (62), (63), (64), (77), (105),\n(107), (108), (110), (122), and (123) of \\cite{Prabhu} are shown to have\nexceptional collections using the Bondal-Uehara Method.\nHence, they admit exceptional collections which are $\\operatorname{Aut}(\\Sigma)$-stable\nand thus provide exceptional collections for the arithmetic forms. Secondly, for the varieties (109), (114), and (115), the set\n$\\mathsf{Frob}(X)$ is a full exceptional collection, which is $G$-stable\nby Lemma~\\ref{lem:FrobNef}. Lastly, Lemma~\\ref{lem:stabprod} guarantees the existence of exceptional collections on products.  Hence, the following varieties also admit stable exceptional collections: (0), (4), (9), (17), (24), (25), (26), (27), (45), (52), (53), (54), (55), (56), (58), (67), (73), (88), (90), (92),\n(93), (97), (103), (111), (112), (113), (118), (119), (120).\n\n\n\n\\subsection{Centrally symmetric toric Fano varieties} \\label{sec:centsym}\n\nPolytopes with the highest degree of symmetry are the \\emph{centrally\nsymmetric} polytopes, i.e., $-P = P$.\nThe smooth split toric varieties $X$ whose anti-canonical polytope is full-dimensional and centrally symmetric were classified in \\cite{VosKly}. It was shown that any such variety (which we refer to as a \\emph{centrally symmetric toric Fano varieties}) is isomorphic to a product of projective lines and \\emph{generalized del Pezzo varieties} $V_n$ of dimension $n = 2m$.  Note that $V_2 = \\mathsf{dP}_6$ and $V_4$ is the missing (116)\nfrom the list in Section~\\ref{sect:4fold} (this is (118) in the enumeration found in \\cite{Batyrev}). The goal of this section is to exhibit full stable exceptional collections on $V_n$, which in turn yields stable exceptional collections for any centrally symmetric toric Fano variety, in light of Lemma~\\ref{lem:stabprod}.\n\nIn \\cite[Theorem 6.6]{CT}, Castravet and Tevelev found $\\operatorname{Aut}(\\Sigma)$-stable full\nstrong exceptional collections for the varieties $V_n$.\nThe authors of this paper had independently discovered the same\nexceptional collection (up to a twist by a line bundle).\nNevertheless, the perspective here may be of independent interest, so we\nsketch the argument. The authors give a more detailed analysis in \\cite{BDMdP}.\n\nThe variety $V_{n}$ with $n= 2m$ has rays given by\n$$\\begin{array}{rl}\ne_1 &= (1,0,\\cdots,0)\\\\\ne_2 &= (0,1,\\cdots,0)\\\\\n&\\vdots\\\\\ne_n &= (0,0,\\cdots,1)\\\\\ne_{n+1} &= (-1,-1,\\cdots,-1)\\\\\n\\end{array}\n\\hspace{.4cm}\n\\begin{array}{rl}\n\\bar{e}_1 &= (-1,0,\\cdots,0)\\\\\n\\bar{e}_2 &= (0,-1,\\cdots,0)\\\\\n&\\vdots\\\\\n\\bar{e}_n &= (0,0,\\cdots,-1)\\\\\n\\bar{e}_{n+1} &= (1,1,\\cdots,1)\n\\end{array}$$ and whose maximal cones are given as follows.  Among the rays $e_1,..., e_{n+1}$, omit a single $e_i$.  From the remaining $n = 2m$ rays, choose $\\frac n 2$ of them and take their antipodes \\cite[Proof of Thm. 5]{VosKly}. Note that $V_2 = \\mathsf{dP}_6$ (whose fan is given in Figure~\\ref{fig:fans}).  The number of maximal cones $c(n)$ of $V_n$ is given by\n\\[\nc(n)=\\frac{(n+1)!}{(\\frac n 2)!^2} = \\frac{(2m +1)! }{m!^2}\\ .\n\\]\nThere's a natural action of $S_{n+1} \\times C_2$, where $S_{n+1}$\npermutes $e_1,\\ldots,e_{n+1}$ and $\\bar{e}_1\\ldots\\bar{e}_{n+1}$ in the\nobvious way.  The $C_2$-action is simply the antipodal map on the\ncocharacter lattice --- we will refer to it as ``the involution.''\nClearly, the involution interchanges $e_i$ and $\\bar{e_i}$. \n\nThe variety $V_n$ is of importance in birational geometry due its\nappearance in the factorization of the standard Cremona transformation\nof $\\mathbb{P}^n$. In fact, as is well-known, $V_n$ can be explicitly obtained\nfrom $\\mathbb{P}^n$ as follows. First blow up the torus fixed points, then\nflip the (strict transforms) of the lines through these points, then\nflip the (strict transforms) of planes through these points, \\ldots, up\nuntil, and not including, the half-dimensional linear subspaces. The\nresulting variety is $V_n$. For more, see \\cite{Casagrande}. \n\nSince $V_n$ and the blow up of $\\mathbb{P}^n$ at its torus fixed points are\nisomorphic in codimension $1$, they have isomorphic Picard groups. We\nuse a basis $H,E_1, \\ldots, E_{n+1}$ for $\\operatorname{Pic}(V_n)$,\nwhich correspond to the hyperplane section and the\nexceptional divisors of the blown up $\\mathbb{P}^n$.\nWe have\n\\[\n[e_i] = E_i, \\quad\n[\\bar{e}_i] = (H-\\sum_{j=1}^{n+1}E_j)+E_j\n\\]\nwhere $S_{n+1}$ permutes the $E_i$ leaving $H$ fixed, and the involution\nis represented by the following matrix\n\\[\n\\begin{pmatrix}\nn & 1 & 1 & \\cdots & 1 \\\\\n1-n & 0 & -1 & \\cdots & -1 \\\\\n1-n & -1 & 0 & \\cdots & -1 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\\n1-n & -1 & -1 & \\cdots & 0 \\\\\n\\end{pmatrix} \\ .\n\\]\n\nFor each $c \\in {\\mathbb Z}$ and $J \\subset \\{1, \\ldots, n+1\\}$, define \n\\[\n F_{c,J} := c\\left(\\sum_{i=1}^{n+1}E_i - H\\right) - \\sum_{j \\in J} E_j .\n\\]\nNote that the involution takes $F_{c,J}$ to $F_{|J|-c,J}$. Then, \n\n\\begin{prop} \\label{prop:Vn}\n The set of $F_{c,J}$ with\n \\begin{enumerate}\n  \\item $\\displaystyle{|J|-\\frac{n}{4} \\le c \\le \\frac{n}{4}}$, or\n  \\item $\\displaystyle{\\frac{n+2}{4} \\le c \\le |J| - \\frac{n+2}{4}}$.\n \\end{enumerate}\nform a full strong $(S_{n+1} \\times C_2)$-stable exceptional collection\non $V_n$ under any ordering of the\nblocks such that $|J|$ is (non-strictly) decreasing.\n\\end{prop}\n\n\\begin{sproof}\nThis collection is the same as that of \\cite[Theorem 6.6]{CT} up to a\ntwist by a line bundle.\nThus, we only sketch an argument here (expanded in \\cite{BDMdP}).\nOne checks that the description of ``forbidden cones'' given by\nBorisov and Hua in \\cite{BH} shows that relevant cohomology groups\nvanish --- this shows that it is a strong exceptional collection.\nTo prove generation, one considers the series of flips required to reach\n$\\mathbb{P}^n$ blown up at $n+1$ points.\nUsing the description of the semi-orthogonal decompositions in\n\\cite{BFK}, the line bundles can be shown to generate the necessary\nadmissible subcategories of each intermediate birational model.\n\\end{sproof}\n\nSince any centrally symmetric toric Fano variety is a product of projective lines and the varieties $V_n$, Lemma~\\ref{lem:stabprod} yields the following:\n\n\\begin{cor}\\label{cor:centsym}\nAny form of a centrally symmetric toric Fano variety admits a full\nstrong exceptional collection consisting of vector bundles.\n\\end{cor}\n\n\\subsection{Toric varieties from the Weyl fans of type A} \\label{sec:X(An)}\n\nOne method for identifying toric varieties with large symmetry groups is to start with root systems. Let $R$ be a root system in a Euclidean space $E$. The ${\\mathbb Z}$-lattice generated by $R$ is denoted $M(R)$, while its dual in $E^\\vee$ is denoted by $N(R)$. For every set $S$ of simple roots in $E$, we have the dual cone corresponding to a closed Weyl chamber\n\\begin{displaymath}\n \\sigma_S := \\{ f \\in E^\\vee \\mid \\langle f, \\alpha \\rangle \\geq 0 \\ , \\ \\forall \\alpha \\in S\\}.\n\\end{displaymath}\nThe cones $\\sigma_S$ are the maximal cones for a fan $\\Sigma_R$ in $E^\\vee$. We denote the associated toric variety by $X(R)$. Recall that an automorphism of $R$ is an element of $\\operatorname{GL}(E)$ preserving $R$. Let $W(R)$ be the Weyl group and $\\Gamma(R)$ the symmetry group of the Dynkin diagram of $R$. It is well-known that\n\\begin{displaymath}\n \\text{Aut}(R) \\simeq W(R) \\rtimes \\Gamma(R).\n\\end{displaymath}\nAny automorphism of $R$ induces an action on the fan $\\Sigma(R)$, which yields a homomorphism $\\phi: \\text{Aut}(R) \\to \\text{Aut}(\\Sigma(R))$.\n\n\\begin{lem} \\label{lem:X(R)FanAut}\n The map $\\phi:  \\operatorname{Aut}(R) \\to \\operatorname{Aut}(\\Sigma(R))$ is an isomorphism.\n\\end{lem}\n\n\\begin{proof}\n First note that the set $R$ can be reconstructed from $\\Sigma(R)$ by taking the union of the extremal rays generating the dual cones $\\sigma_S^\\vee$ for all $\\sigma_S$. Thus any symmetry of the fan induces a symmetry of $R$. This gives the inverse map to $\\phi$.\n\\end{proof}\n\nHere we focus on the case $R = A_n$. In \\cite{LosevManin}, the authors showed that $X(A_n)$ is a moduli space of rational curves with $(n+1)$ marked points and $2$ poles. Another useful proof appeared in \\cite{BatyrevBlume}. \n\nUsing this perspective, Castravet and Tevelev exhibited an exceptional\ncollection on $X(A_n)$ that is stable under the action of permuting the\nmarked points and flipping the poles, i.e., an $(S_{n+1} \\rtimes\nC_2)$-stable collection. Here we demonstrate that Castravet and\nTevelev's exceptional collection satisfies the conditions of\nProposition~\\ref{prop:blackbox} and hence descends to an exceptional\ncollection on any form of $X(A_n)$ (in characteristic $0$). \n\nTo do this requires a bit of translating divisors and actions from the moduli-theoretic language to the toric language. We recall the moduli-theoretic languge. \n\n\\begin{defn}\n Let $N$ be a set of order $n$. A \\emph{chain of polar} ${\\mathbb P}^1$'s is a $\\left(\\{0,\\infty\\} \\cup N\\right)$-marked linear nodal chain of $\\mathbb{P}^1$'s with $0$ on the left tail and $\\infty$ on the right tail. A chain of polar ${\\mathbb P}^1$'s is \\emph{stable} if \n \\begin{enumerate}\n  \\item marked points do not coincide with nodes,\n  \\item only $N$-marked points are allowed to coincide,\n  \\item each component of the chain has at least three special points (nodes or marked points). \n \\end{enumerate}\n We write $LM_N$ for the corresponding moduli space. We also use $LM_n$ depending on the context.  Note that the universal curve over $LM_n$ is isomorphic to $LM_{n+1}$.\n\\end{defn}\n\n\n\\begin{thm}\\label{thm:univcurve}\n The toric variety $X(A_{n-1})$ is isomorphic to $LM_n$. Moreover, if we fix an embedding $A_{n-1} \\to A_n$, the corresponding map $X(A_n) \\to X(A_{n-1})$ is the universal curve.\nMoreover, $X(A_n) \\to X(A_{n-1})$ is a toric morphism.\n\\end{thm}\n\n\\begin{proof}\nThis is \\cite[Theorem 2.6.3]{LosevManin}. See also \\cite[Theorem 3.19]{BatyrevBlume}. \nThe map is consequently toric by \\cite[Proposition~1.4]{BatyrevBlume}.\n\\end{proof}\n\nUnder this isomorphism, the closures of the torus orbits on $X(A_n)$ have the\nfollowing moduli-theoretic description.\nFix a partition $N_1 \\sqcup N_2 = N$ and let $\\delta_{N_1}$ denote the\ndivisor parametrizing polar chains of length exactly $2$ having the\nfirst marked by $N_1$ and the last marked by $N_2$.\nFor a partition with more parts, $N_1 \\sqcup N_2 \\sqcup \\cdots \\sqcup N_t = N$,\none has the locus $Z_{N_1,\\dots,N_t}$ parametrizing polar chains of length exactly $t$,\nwhere the $i$-th ${\\mathbb P}^1$ is marked by $N_i$.\nThese loci are precisely the proper torus orbit closures on $X(A_n)$.\n\nNote that each loci is a complete intersection\n\\begin{displaymath}\n Z_{N_1,\\dots,N_t} := \\delta_{N_1} \\cap \\delta_{N_1 \\cup N_2} \\cap \\cdots \\cap \\delta_{N_1 \\cup \\cdots \\cup N_{t-1}}. \n\\end{displaymath}\nMoreover, we have an isomorphism \n\\begin{displaymath}\n Z_{N_1,\\dots,N_t} \\simeq LM_{N_1} \\times LM_{N_2} \\times \\cdots \\times LM_{N_t}\n\\end{displaymath}\nwhere the left node of each ${\\mathbb P}^1$ is marked with $0$ and the right node is marked with $\\infty$.\nThus, we have toric morphisms \n\\begin{displaymath}\n  i_{N_1,\\ldots,N_t} : LM_{N_1} \\times LM_{N_2} \\times \\cdots \\times LM_{N_t} \\to LM_N \\ .\n\\end{displaymath} \nAlso, for each subset $K \\subset N$, we get a forgetful map $\\pi_K: LM_N\n\\to LM_K$, which is a toric morphism since it is a composition of maps\nfrom Theorem~\\ref{thm:univcurve}.\n\nRecall there is a set of line bundles $\\mathbb{G}_N$ on $LM_N$ \\cite[Definition 1.5]{CT}, and one generates a larger set $\\mathsf{H}_N$ of sheaves via\n\\begin{displaymath}\n \\mathsf{H}_N := \\left \\lbrace \\left(i_{N_1,\\ldots,N_t}\\right)_*\n(G_{l_1} \\boxtimes \\ldots \\boxtimes G_{l_t})\n\\mid \\forall N_1 \\cup \\cdots \\cup N_t = N \\ , \\ G_{l_j}\n\\in \\mathbb{G}_{N_j} \\right \\rbrace,\n\\end{displaymath}\nwhere $i_{N_1,\\ldots,N_t}: Z_{N_1, \\ldots, N_t} \\hookrightarrow LM_N$ is the inclusion. \n\n\\begin{thm}\\label{thm:CTexcp}\n There is an ordering on the set \n \\begin{displaymath}\n  \\mathsf{CT}_N :=\n\\mathsf{H}_N \\cup \\left(\n\\bigcup_{K \\subsetneq N} \\{ \\pi_K^\\ast E \\mid \\ E \\in \\mathsf{H}_K \\} \\right)\n\\cup \\{ \\mathcal{O} \\}\n \\end{displaymath}\nmaking it into an $(S_N \\rtimes C_2)$-stable exceptional collection under permutations of the two sets of markings.\n\\end{thm}\n\n\\begin{proof}\n This is \\cite[Proposition 1.5]{CT}.\n\\end{proof}\n\n\\begin{prop}\\label{prop:CTaut=rootAut}\n The action of $S_{n+1} \\rtimes C_2$ given by permuting the two sets of marked points corresponds to the action of $\\operatorname{Aut}(A_n)$ on $X(A_n)$. \n\\end{prop}\n\n\\begin{proof}\n We use the standard presentation of the root system for $A_n$ as $e_i - e_j$ for $1 \\leq i < j \\leq n+1$ and follow \\cite[Construction 3.6]{BatyrevBlume}. The embedding $A_n \\hookrightarrow A_{n+1}$ gives the universal curve $X(A_{n+1}) \\to X(A_n)$. For $i \\in \\{1,\\ldots,n\\}$, we take the $(n+1)$ projections $A_{n+1} \\to A_n$, whose kernels are generated by $e_i - e_{n+1}$ for $1 \\leq i \\leq n+1$. These give sections $s_i : X(A_n) \\to X(A_{n+1})$. Finally, for the polar sections, we have the dual vector $v_{n+2}$. The vectors $v_{n+2}$ and $-v_{n+2}$ give toric invariant divisors which are isomorphic to $X(A_n)$ \\cite[Proposition 1.9]{BatyrevBlume}. The isomorphisms give the other sections $s_0$ and $s_{\\infty}$. \n \n The Weyl group is the permutation group of the $e_i$, and hence of the $e_i - e_{n+2}$. In particular, it permutes the $s_i$. The outer involution acts on the fan by negation and thus exchanges the cone corresponding to $v_{n+2}$ with the cone corresponding to $-v_{n+2}$.\n\\end{proof}\n\n\\begin{cor}\\label{cor:CTisstable}\n The set $\\mathsf{CT}_N$ is $\\operatorname{Aut}(\\Sigma(A_n))$-stable. \n\\end{cor}\n\n\\begin{proof}\n This is an immediate corollary of Lemma~\\ref{lem:X(R)FanAut} and Proposition~\\ref{prop:CTaut=rootAut}. \n\\end{proof}\n\n\\begin{prop}\\label{prop:CTistoric}\nEach object in the collection $\\mathsf{CT}_N$ is torus-equivariant. \n\\end{prop}\n\n\\begin{proof}\nLine bundles are always isomorphic to torus-equivariant line bundles,\nso all objects in $\\mathbb{G}_N$ are torus-equivariant.\nThere is a canonical equivariant structure on tensor products and on\npullbacks by equivariant morphisms (see \\S{2}~of~\\cite{BFK2});\nthus each object $G_1 \\boxtimes \\ldots \\boxtimes G_n$\nis torus-equivariant for $G_{l_j} \\in \\mathbb{G}_{N_j}$.\nLet $i : Z \\to X$ be shorthand for some map $i_{N_1,\\ldots,N_t}$.\nThere is a splitting of tori $T = S \\times S'$ where $Z$ is an $S$-toric\nvariety and $S'$ acts trivially on $i(Z)$.\nLet $\\psi : T \\to S$ denote the projection. \nWe have a composition of functors\n\\[\n\\mathsf{D^b}(\\operatorname{Coh}_S Z) \\to\n\\mathsf{D^b}(\\operatorname{Coh}_T Z) \\to\n\\mathsf{D^b}(\\operatorname{Coh}_T X)\n\\]\nwhere the first map is the functor $\\operatorname{Res}_\\psi$\n(\\S{2.9}~of~\\cite{BFK2}) and the second map is the $T$-equivariant\npushforward (\\S{2.5}~of~\\cite{BFK2}).\nThis composition reduces to the ordinary pushforward\n$i_\\ast : \\mathsf{D^b}(Z) \\to \\mathsf{D^b}(X)$ when the equivariant\nstructure is forgotten.  We conclude that each object of\n$\\mathsf{H}_K$ is torus-equivariant\nand the result follows.\n\\end{proof}\n\nWe now prove the main result of this section:\n\n\\begin{prop}\\label{prop:X(An)excpcoll}\n Let $k$ be a field of characteristic zero and $X$ a form of $X(A_n)$ over $k$. Then $X$ admits a full exceptional collection of sheaves. \n\\end{prop}\n\n\\begin{proof}\n Combining Theorem~\\ref{thm:CTexcp}, Corollary~\\ref{cor:CTisstable}, and\nProposition~\\ref{prop:CTistoric} allows us to appeal to\nProposition~\\ref{prop:blackbox} and conclude that $\\mathsf{CT}_N$\ndescends to an exceptional collection of sheaves on $X$. \n\\end{proof}\n\n\\begin{rem}\n To remove the characteristic zero assumption one needs to extend generation results of \\cite{CT} to nonzero characteristic. This could conceivably be done by reversing the flow of reasoning in \\cite{CT}, using the fact we know the collections for $V_n$ in any characteristic. We do not pursue this. \n\\end{rem}\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{} \n\nGalaxy bimodality, described by the red sequence and blue cloud, has been central to our understanding of galaxy evolution since the turn of this century \\citep[e.g.,][]{2004ApJ...600..681B,2004ApJ...608..752B,2009ARA&A..47..159B}. Passive galaxies follow a narrow color-magnitude relation while star-forming galaxies in the blue cloud have a broader range of optical colors, resulting from a range of stellar populations, star formation rates and dust obscuration. Although star-forming galaxies are diverse, they do fall along well-established correlations with mass, including the mass-metallicity relation \\citep[e.g.,][]{2004ApJ...613..898T}, star formation rate versus mass \\citep[e.g.,][]{2007ApJ...660L..43N} and declining dust content with decreasing mass \\citep[e.g.,][]{2006ApJ...639..157W}. These relations manifest themselves in broadband photometry, albeit outside the optical wavelength range, as illustrated by the dependence of infrared colors on galaxy type \\citep[e.g.,][]{2011ApJ...735..112J}.\n\nThe far-ultraviolet and mid-infrared are both star formation rate tracers, with the former tracing massive stars while the latter traces blackbody emission from warm dust. While far-ultraviolet luminosity is directly proportional to star formation rate, for $\\lesssim L^*$ galaxies mid-infrared luminosity is proportional to star formation rate to the power of $\\sim 1.3$ \\citep[e.g.,][]{2015AA...584A..87C,2017ApJ...847..136B}, which is a consequence of dust content varying with galaxy mass. We thus expect a blue sequence to be present in ultraviolet-infrared color-magnitude diagrams.\n\nTo measure the ultraviolet-infrared color-magnitude relation, we use the local galaxy sample of \\citet{2014ApJS..212...18B, 2017ApJ...847..136B} and their multiwavelength matched aperture photometry (in AB magnitudes). We limit the sample to galaxies with $m_{W2}-m_{W3}>-0.5$, which excludes passive galaxies from the \\citet{2014ApJS..212...18B} sample, and we remove active galactic nuclei with the emission line ratio criterion of \\citet{2003MNRAS.346.1055K}. To correct the GALEX $FUV$ photometry for internal dust obscuration we use $A_{FUV}  \\propto (M_{FUV} - M_{NUV})$, leaving the constant as a free parameter that we use to minimize the scatter of the color-magnitude relation. \n\nIn Figure~\\ref{fig:thefigure} we present the ultraviolet-infrared color-magnitude plot of $z\\sim 0$ star-forming galaxies, using the $FUV$, $NUV$, and WISE $W3$ photometry. We find the best relation is produced when $A_{FUV}  = 2.6 (M_{FUV} - M_{NUV})$, which is shallower than the dust extinction relation of \\citet{2011ApJ...741..124H}, where $A_{FUV}  = (3.83\\pm 0.48) [M_{FUV} - M_{NUV} - (0.022\\pm 0.024)]$. As a cross check of our results, in Figure~\\ref{fig:thefigure} we also plot photometry of $z<0.05$ GAMA galaxies \\citep{2016MNRAS.460..765W} with WISE $m_{W2}-m_{W3}>-0.5$, and we find good agreement although GAMA spans a smaller range of $M_{W3}$ than \\citet{2014ApJS..212...18B, 2017ApJ...847..136B}. We also observe similar relations when we replace WISE $W3$ with WISE $W4$ or {\\it Spitzer} $24~{\\rm \\mu m}$, albeit with more scatter. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth,angle=0]{blueseqplot.pdf}\n\\includegraphics[width=0.49\\textwidth,angle=0]{gH.pdf}\n\\caption{The ultraviolet-infrared (left) and $g-H$ (right) color-magnitude relations for star-forming galaxies. Some of the outliers in the $g-H$ are labelled, and these are often merging galaxies rather than spirals. As the \\citet{2014ApJS..212...18B,2017ApJ...847..136B} sample deliberately selected galaxies to span parameter space, it shows more scatter than the magnitude limited GAMA sample.  \\label{fig:thefigure}}\n\\end{center}\n\\end{figure}\n\nThe best-fit color-magnitude relation is given by \n\\begin{equation}\nM_{W3} = -14.8 - 2.1 \\times \\left[ 2.6 (M_{FUV} - M_{NUV}) - M_{W3} \\right].\n\\end{equation}\nUsing the  \\citet{2014ApJS..212...18B, 2017ApJ...847..136B} sample, we find the $1\\sigma$ scatter of $M_{W3}$ about the relation is $\\sigma_{W3} =1.6~{\\rm mag}$. If ultraviolet - infrared color was used as a distance indicator then the 68\\% scatter of the distance would be a factor of $\\sim 2$. \n\nWe note color-magnitude relations for blue galaxies have been identified previously, including the median optical color of blue galaxies varying with magnitude \\citep{2004ApJ...600..681B}. Furthermore, \\citet{1982ApJ...257..527T} identified a tight $B-H$ color-magnitude relation for spiral galaxies, and in the right panel of Figure~\\ref{fig:thefigure} we reproduce this relation for star-forming galaxies using SDSS $g$ and 2MASS $H$-band photometry.  \\citet{1982ApJ...257..527T} attributed this relation to specific star formation rate, chemical abundances and\/or initial mass function varying with mass. Interestingly, we do see some outliers in the $g-H$ versus $M_H$ diagram, including merging starbursts. These outliers are not unexpected, given $g$ and $H$ trace different galaxy properties, whereas the ultraviolet and mid-infrared are both (primarily) star formation rate tracers. \n\nIn this note we have identified and characterized the ultraviolet-infrared color-magnitude relation of star-forming galaxies. The ultraviolet to mid-infrared flux ratios of star-forming galaxies span over two orders of magnitude and show a clear dependence on absolute magnitude from $M_{W3}\\sim -13$ to $M_{W3}\\sim -25$, which may present problems for models of galaxy spectral energy distributions that have been largely verified on $\\sim L^*$ galaxies. The color-magnitude relation of star-forming galaxies illustrates the (broadband) spectral diversity of star-forming galaxies that results from established correlations between the physical properties and mass, including the mass-metallicity relation.\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Background}\n\\label{sec:introduction}\n\nConventional dynamic random-access memory (DRAM)\nscaling has reached a critical tipping point as the miniaturization of the DRAM cell has plateaued in recent years. Feature size scaling below the 20 $\\text{nm}$ technology node is met with numerous challenges such as\nshorter retention times, higher leakage currents, and increased fault rates~\\cite{park2015technology}.\nSolutions to address these concerns include improved DRAM fault detection and recovery~\\cite{wang2017improving}, as well as architectural techniques to enhance DRAM scaling~\\cite{kim2015architectural}. \n\nA promising solution to the memory scaling problem is to realize the main memory system using non-volatile technologies~\\cite{mutlu2015main}. \nExamples of emerging non-volatile memories (NVMs) include spin-transfer torque magnetoresistive random-access memory (STT-MRAM), ferroelectric random-access memory (FeRAM), resistive random-access memory (ReRAM), and phase-change memory (PCM).\nInterest in the commercial application of such NVMs has increased significantly. For instance, Intel's current line of 3D XPoint memory systems utilize PCM-based NVM technology~\\cite{wyrwas2017proton}, and IBM and Everspin's solid-state drive comes with STT-MRAM write caches~\\cite{everspin}. While NVMs offer attractive features, such as high density, low leakage, and non-volatile data retention, they also suffer from poor endurance and high access latency in their current implementation.\n\nMemory security has come under more scrutiny over the years.\nThis is because of attacks such as \\textit{Spectre}~\\cite{kocher2019spectre} and \\textit{Meltdown}~\\cite{lipp2018meltdown}, which targets the side-channels associated with speculative execution and out-of-order execution, respectively, have exposed severe vulnerabilities in a wide array of currently deployed processors and their memory architectures. \nIn the case of NVMs, data remanence after power-down\npresents a severe threat to data confidentiality, as attackers aiming to steal private data can do so easily by mounting cold-boot attacks~\\cite{halderman2009lest} or other removal attacks like stealing the memory module (DIMM)~\\cite{young2015deuce}. \nMoreover, magnetic memories like STT-MRAM are highly sensitive to stray magnetic fields. \nAs such, magnetic field-based attacks~\\cite{jang2015self} can be used to corrupt the stored data or compromise the memory's functional integrity, resulting in a denial-of-service (DoS) attack. \nHence, such security vulnerabilities pose a significant impediment to the pervasive and large-scale proliferation of NVMs in the memory industry. \n\n\\subsection{Related work in Memory Security}\nPrior works on securing NVMs have focused mainly on memory encryption schemes, which are necessary to prevent attackers from exploiting data remanence in the off-state. \nChhabra \\textit{et al.} proposed an incremental encryption scheme~\\cite{chhabra2011nvmm} for NVMs where only inert memory pages, which have not been accessed for several clock cycles, are encrypted selectively. \nThe working set of the memory (which is in current use) is in plaintext and, hence, incurs no encryption overhead on access. Such a selective encryption ensures that the majority of the main memory content (but not all) remains encrypted at all times, without overly compromising the performance. \nHowever, it requires dedicated hardware, inert page prediction, and scheduling for its implementation. \nA sneak-path encryption (SPE) scheme was demonstrated for memristor-based NVMs in~\\cite{kannan2014secure}, wherein sneak paths in the memristor crossbar array are exploited to apply encryption pulses to change the resistances of the memory cells, and hence, encrypt the stored data. \n\nIn~\\cite{young2015deuce}, the authors proposed DEUCE, a dual counter encryption for PCM memories, which significantly reduces the number of modified bits per writeback, to improve performance and lifetime of the memory. \nThis scheme aims to mitigate the impact of the avalanche effect~\\cite{mandal2012performance} occurring during memory encryption, by re-encrypting and writing back only the modified words during any write operation. Swami~\\textit{et al.} took this concept forward and proposed SECRET~\\cite{swami2016secret}, a smart encryption scheme for NVMs, which integrates word-level re-encryption and zero-based partial writes to reduce memory write operations. They also demonstrate write optimization through the use of\n``energy masks'' (i.e., bit templates XORed with ciphertext to obtain lower energy dissipation)\nin the encryption XOR logic, which minimizes the bit flips in the encryption process, thereby reducing the total write energy.\nAn advanced counter-mode encryption (ACME) was presented in~\\cite{swami2018acme}, which utilizes the write leveling architecture inherent in PCM memories, to perform counter-write leveling. \nACME helps to avoid \\textit{Rowhammer}-type attacks by preventing the counter associated with any single cache line from overflowing.\n\nThe impact of contactless tampering on STT-MRAMs using external magnetic fields was highlighted in~\\cite{jang2015self}. \nUsing micromagnetic simulations, the authors of~\\cite{jang2015self}\nshowed how magnetic field-based attacks could corrupt the contents of STT-MRAM cells. Techniques to protect against contactless attacks proposed in~\\cite{jang2015self} included (i) an on-chip sensor to detect magnetic field-based incursions, and (ii) error correction modules to compensate cell failures arising due to magnetic field attacks. However, these techniques incur large energy and area penalties due to the additional hardware imposed by the magnetic field sensor and the error correction scheme.\n\n\\subsection{Contributions}\nIn this paper, we present an alternative to conventional NVMs such as STT-MRAM and PCM, in the form of \\textit{SMART: A Secure Magnetoelectric Antiferromagnet-Based Tamper-Proof Non-Volatile Memory}. \nSMART memory leverages the room-temperature linear magnetoelectric (ME) effect in antiferromagnets (AFMs) like chromia~\\cite{rado1961observation}, which can be switched solely using voltage pulses, without the use of electric currents, leading to ultra-low energy ($\\sim$ pico-Joules) operation. \nFurther, the intrinsic dynamics of AFMs is typically in the terahertz regime ($\\sim 10^{12}$ Hz)~\\cite{khymyn2017antiferromagnetic}, which could enable picosecond time-scale reversal of the AFM domain. \nIn addition to its energy and latency benefits, SMART memory offers a significant advancement in terms of secure and tamper-proof data storage. \nFor example, AFMs do not exhibit a magnetic signature since they do not have a net external magnetic moment, unlike ferromagnets (FM). \nHence, the SMART memory cannot be probed or switched with external magnetic fields, unlike the way STT-MRAMs can. \nThis, in turn, eliminates the possibility of magnetic field attacks undermining data integrity or aiming to induce DoS. \nTo address the post-shutdown data remanence of SMART memory, we demonstrate an in-memory encryption scheme employing  ME-AFM transistor-based controlled-NOT (CNOT) logic. \nWe discuss the resilience of the SMART memory against attacks aiming to undermine data confidentiality and data fidelity, in both powered-on and powered-off states. \nThe main contributions of this work can be summarized as follows:\n\n\\begin{enumerate}\n    \n\\item We discuss the design of SMART, a secure ME-AFM-based NVM and implement its SPICE circuit model to simulate the memory performance. \n    \n\\item We demonstrate the resilience of SMART memory against magnetic field and temperature attacks, which can affect other NVMs like STT-MRAM. We explore the implications of various side-channel attacks on the SMART memory.\n    \n\\item We present an in-memory encryption scheme with ME-AFM transistor-based CNOT gates, called \\textit{Memcryption}, to protect the data stored in SMART memory against cold-boot and stolen DIMM attacks, while incurring low encryption latency overheads.\nWe like to mention here that \\textit{Memcryption} is specifically tailored for the ME-AFMRAM, not for a generic NVM. Also, it does not secure the memory system against \\textit{bus snooping} attacks; such attacks are beyond the scope of this work.\n\n\\end{enumerate}\n\nIn the next section, we describe the modeling, implementation and benchmarking of the proposed ME-AFM memory both at cell- and array-level, before proceeding to evaluate its security properties in Section~\\ref{sec:security}.\n\n\\section{Device model and functionality}\n\\label{Modeling}\n\\subsection{The magnetoelectric effect}\nThe linear ME effect~\\cite{agyei1990linear} represents the coupling between applied magnetic field and induced polarization or between applied electric field and induced magnetization in non-centrosymmetric crystals like chromia ($\\text{Cr}_2\\text{O}_3$). Compared to the STT-based magnetization reversal of FMs requiring electric currents on the order of $\\sim10^6$ A\/cm$^2$ and incurring associated Joule heating, the ME effect provides an energy-efficient, all-electrical switching of the roughness-insensitive boundary magnetization of chromia~\\cite{echtenkamp2013electric}. Additionally, chromia is an AFM; hence, the net bulk magnetic moment\n(i.e., the difference of the sublattice magnetization vectors) vanishes and becomes imperceptible externally. \nHowever, the boundary magnetization is strongly coupled to the AFM order parameter. That is, the electrical switching of the AFM order results in reversal of the boundary magnetization~\\cite{wu2011imaging}, which is used to encode the information in ME-AFM memories.\n\nThe uncompensated surface moments at the (0001) surface of chromia result in an equilibrium boundary magnetization, which could be in one of the two oppositely aligned,\ndegenerate domain states. \nThe degeneracy between the domains is lifted through ME annealing, which allows the preferential selection of one of the states~\\cite{he2010robust}. That is, the ME annealing polarizes the surface and results in\na single-domain surface moment. \nIsothermal switching between these single domain states using an electric field $E$ and a small, symmetry-breaking\nDC magnetic field $H$ has been demonstrated experimentally~\\cite{he2010robust, fallarino2015magnetic}. \nThe critical condition for such ME switching is that the magnitude of the $E\\cdot H$ product must exceed the ME threshold energy barrier, which was shown experimentally to be as low as $\\approx$ 1 J\/m$^3$~\\cite{brown1969domain, martin1966antiferromagnetic}.\n\n\\subsection{ME-AFMRAM : Working principle}\nThe chromia-based ME-AFMRAM, which is at the heart of our SMART memory, is shown in Fig.~\\ref{fig:AFMRAM}. Experimentally demonstrated by Kosub \\textit{et al.}~\\cite{kosub2017purely}, the ME-AFMRAM has a bottom gate electrode (Platinum gate in the figure) for applying the gate voltage $V_G$ and providing the necessary electric field to write data into the memory. A small, symmetry-breaking magnetic field ($\\approx$ 30 mT) is provided by the stray field of a permanent magnet. A positive voltage $V_G$ will orient the bulk order and, hence, put the surface magnetization in one domain (with surface moments pointing up), whereas a negative voltage will result in the surface magnetization relaxing to the opposite domain (with surface moments pointing down). These two states correspond to binary levels `1' ($V_G > 0$) and `0' ($V_G < 0$), respectively. A gate voltage of 0 V corresponds to the `hold' mode of the memory cell. Note that the cell serves as non-volatile memory in all gate-voltage ranges, not only for $V_G = 0$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.28]{figures\/AFMRAM.pdf}\n\\caption{Chromia-based magnetoelectric antiferromagnetic random-access memory. Data (1\/0) is written by applying a voltage ($+\/-$) to the bottom gate electrode. Read-out is achieved using an anomalous Hall bar electrode placed on top, by applying a Hall bias.}\n\\label{fig:AFMRAM}\n\\end{figure}\n\nThe read-out is achieved using an anomalous Hall (AH) bar electrode setup, which discerns the boundary magnetization of chromia by sensing the proximity effect-induced magnetization in the nearby Platinum (Pt) electrode, thereby producing a \nproportional Hall voltage $V_{\\text{xy}}$ (or $V_{\\text{AHE}}$)~\\cite{kosub2015all}. Traditionally, the order parameter of AFMs is read-out via an exchange bias arrangement~\\cite{toyoki2015magnetoelectric} in another FM attached adjacently to the AFM surface. However, the exchange bias and the FM's hysteresis increase the coercive voltage required to overcome the ME barrier and, hence, impact the write energy negatively. To avoid this effect, Kosub \\textit{et al.}~\\cite{kosub2017purely} proposed the use of an exclusively ME-AFM setup with an AH read-out of the surface magnetization, thereby eliminating the need for an FM.\nAt the time of writing this paper, a complete physical understanding of the read-out mechanism for the boundary magnetization in chromia is lacking. While the authors in~\\cite{kosub2017purely} have considered an AH-based read-out in their device, recent experiments by C. Binek's group at the University of Nebraska-Lincoln have revealed the contribution of spin-Hall magnetoresistance (SMR) to the read-out signal, which is currently being investigated.\nHowever, note that the magnitude of the signal levels is the same in both cases (AH versus SMR) and also the circuit models developed would remain the same, though with different input parameters. \nFor the purposes of this paper, we consider that the read-out signal is due to the AH effect in the proximal heavy metal, as also discussed in prior experimental work.\n\n\\subsection{Performance modeling}\nThe ME reversal mechanism in chromia can be classified broadly into two categories, depending on the size of the film compared to the characteristic domain-wall (DW) width. For chromia, the typical DW width $\\lambda =\\sqrt{A\/\\mathcal{K}}\\sim$ 50-100 nm, where $A$ is the exchange stiffness constant and $\\mathcal{K}$ is the uniaxial anisotropy energy~\\cite{belashchenko2016magnetoelectric}. \nIf the sample is much smaller than the DW width, the sample reverses via coherent rotation upon application of the ME pressure. For sample dimension comparable to the DW width, ME reversal occurs via DW nucleation and propagation, which is an incoherent switching process.\nFor both coherent rotation and DW propagation, the reversal could be thermally activated for applied ME pressure lower than the energy barrier between the stable domain states. Otherwise, the domain reversal proceeds in the `flow' regime~\\cite{parthasarathy2019dynamics}.\nME-AFMRAM devices currently fabricated have dimensions in the $\\mu$m range, rendering DW propagation the favorable ME reversal mechanism. To characterize the functionality and performance of chromia ME-AFMRAM, we develop circuit models that represent DW-based reversal in both the thermally activated and the flow regimes. We also provide perspectives and future\npotential concerning dimensional scaling of the device, which could enable ultra-fast, coherent, rotation-based reversal. \n\n\\subsubsection{DW reversal of chromia ME-AFMRAM}\nConsider a chromia sample, where the applied ME pressure creates a pressure difference of $\\mathcal{F} =|2\\alpha_{\\text{ME}} E H|$ between the two domains. Here, $\\alpha_\\mathrm{ME}$ is the linear ME coefficient.\n\nIf $\\mathcal{F}> \\mathcal{F}_d$ (i.e., for DW de-pinning pressure), the DW propagates as a viscous flow with velocity given as~\\cite{parthasarathy2019dynamics}\n$$\\nu_{\\text{flow}} = \\frac{\\alpha_{\\text{G}}\\gamma \\lambda}{\\alpha_+\\xi^2}\\Big( \\frac{\\mathcal{F}-\\mathcal{F}_{\\text{d}}}{M_{\\text{s}}} \\Big),$$\nwhere $\\alpha_{\\text{G}}$ is the Gilbert damping constant, $\\gamma$ is the gyromagnetic ratio of electron, $M_{\\text{s}}$ is the sublattice saturation magnetization, and $\\xi=\\frac{\\alpha_{\\text{ME}}E}{\\mu_0 M_{\\text{s}}}$. \nFor a mean free path of $l$ of the DW, the time-scale of ME reversal due to viscous DW propagation is $\\tau_{\\text{flow}}=l\/\\nu_{\\text{flow}}$.\n\nIf $\\mathcal{F}<\\mathcal{F}_{\\text{d}}$, the DW undergoes thermal creep to overcome the de-pinning barrier, with a time-scale~\\cite{parthasarathy2019dynamics}\n$$\\tau_{\\text{creep}}=\\sqrt{\\frac{\\sigma \\mathcal{S}^3}{kT}}\\Big(\\frac{\\mathcal{F}_{\\text{d}}-\\mathcal{F}}{2\\pi\\epsilon}\\Big)\\exp\\Big[{\\frac{\\mathcal{S}^2(\\mathcal{F}_{\\text{d}}-\\mathcal{F})^2}{4\\pi kT\\epsilon}\\Big]},$$\nwhere $kT$ is the thermal energy (25 meV at 300 K), $\\epsilon$, $\\sigma$, and $\\mathcal{S}$\nare the energy, areal density, and surface area, respectively, of the DW. The DW de-pinning pressure is determined by the DW energy, its surface area, and the radius of the non-magnetic de-pinning center.\n\nTo write `1' (`0') into the memory cell, a positive (negative) electric field, $E_\\mathrm{app}$, with a magnitude greater than the critical electric field, $E_\\mathrm{crit}$, is required, in order to meet the DW propagation criteria of $\\mathcal{F}>\\mathcal{F}_d$. In this case, the time to write data into the memory is equal to $\\tau_\\mathrm{flow}$. When $E_\\mathrm{app}$ is less than $E_\\mathrm{crit}$ (i.e., $\\mathcal{F}<\\mathcal{F}_d$), the memory cell is in the hold mode and the retention time is specified by $\\tau_\\mathrm{creep}$. For typical parameters of chromia, we find $\\tau_\\mathrm{creep}\\gg \\tau_\\mathrm{flow}$, which ensures that the memory cell is thermally stable when it is not accessed. Here, the stability of the cell is determined by $\\tau_{\\text{creep}}$, since longer data retention requires the time constant in the hold mode to be larger. The retention time of the cell can be further improved by enlarging the cell dimensions.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.19]{figures\/chromia_circuit_final.pdf}\n\\caption{Equivalent circuit for the chromia ME-AFMRAM cell. $I_{\\text{int}}$, derived from the bit line, writes data on to the node $V_{\\text{ME}}$. The time constant of the write operation is $\\tau_{\\text{flow}}$ ($\\tau_{\\text{creep}}$) if the applied voltage is greater (smaller) than the critical voltage. Read-out is achieved through an AH setup, modeled with a voltage-controlled voltage source. $\\text{C}_\\text{EL}$ is the electrostatic capacitance of the chromia dielectric.}\n\\label{fig:chromia_RC}\n\\end{figure*}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.37]{figures\/MERAM_transient.pdf}\n\\caption{Transient simulations showing write operations on the chromia ME-AFMRAM cell. Note that for writing a `1' the write pulse is positive, and for writing a `0' the write pulse is negative. In this simulation, a series of `1's (0.3 V) and `0's (-0.3 V) are being written to the cell, and then finally `0' is retained once Write Enable is switched off.}\n\\label{fig:AFMRAM_timing}\n\\end{figure}\n\nWe construct a SPICE circuit model to functionally capture the ME reversal dynamics of chromia. The time constant for reversal of the magnetization of chromia due to an applied ME pressure is represented as $R_{\\text{eq}}\\times C_{\\text{eq}}$. Without loss of generality, the circuit model uses $R_{\\text{eq}} = 1$ $\\Omega$, while $C_{\\text{eq}}$ is either $\\tau_{\\text{flow}}$ or $\\tau_{\\text{creep}}$. \nTo construct the full ME-AFMRAM cell, we combine the RC model of the ME response of chromia with the peripheral read\/write circuitry in \\textit{Cadence Virtuoso} using the 15-nm CMOS FreePDK technology. Figure~\\ref{fig:chromia_RC} shows the equivalent circuit of the ME-AFMRAM cell. \nThe write pulse, used to charge the chromia dielectric and switch its\nmagnetization $M$, is provided through the current source $I_{\\text{int}}$ (derived from the bit line) in the write setup. \nFor parameters of chromia listed in Table~\\ref{tab:params}, \n$C_{\\text{flow}}=\\tau_{\\text{flow}}\\sim0.223$ nF, $C_{\\text{creep}}=\\tau_{\\text{creep}}\\sim1$ mF, and $V_{\\text{crit}}= 0.2$ V. For $|V_G| > 0.2$ V, $V_{\\text{ME}}$ tracks $V_G$ and data is written into the cell after a write access latency of $\\tau_\\mathrm{flow}$. \nWhen $|V_G| = 0$ V, data is retained for a time interval of $\\tau_{\\text{creep}}$. Since $\\tau_{\\text{creep}}$ is very large, the response in retention\/creep mode is extremely slow as compared to write\/flow mode. The transient response of the ME-AFMRAM cell is shown in Fig.~\\ref{fig:AFMRAM_timing}, to highlight the write operation. The write latency of the ME-AFMRAM cell is obtained as $\\sim 0.63$ ns, and the energy-per-bit for one write operation is $\\sim 0.063$ pJ, including the energy required to charge the electrostatic capacitance of chromia.\nGiven relative dielectric permittivity of 11 and dimensions noted in Table~\\ref{tab:params}, the electrostatic capacitance of chromia is calculated as $5.8$ aF.\n\n\\subsubsection{Anomalous Hall read-out}\nTo evaluate the read cycle, we set the signals WE to 0 and RE to 1 in Fig.~\\ref{fig:chromia_RC}. The read setup is designed to sense the boundary magnetization of chromia through an AH arrangement, which transduces the magnetization into a voltage signal. This transduction process is modeled using a voltage-controlled voltage source\n(VCVS). Typically, a heavy metal such as Pt is used to sense the proximity effect-induced moment from the coupled chromia layer~\\cite{kosub2017purely}.\n\nThe AH voltage sensed from the Hall bar arrangement is given as~\\cite{griffiths2017anomalous}\n$$V_{\\text{AHE}}=\\Big(\\frac{\\mu_0R_{\\text{s}}}{t_{\\text{Hall}}}I_{\\text{Hall}}\\Big)M_{\\text{z}},$$\nwhere $\\mu_0$ is the vacuum permeability, $R_{\\text{s}}$ is the AH coefficient, $I_{\\text{Hall}}$ is the Hall bias current, $t_{\\text{Hall}}$ is the thickness of the Hall layer and $M_{\\text{z}}$ is the proximity effect-induced magnetization. In the case of Pt\/Cr$_2$O$_3$, $R_{\\text{s}}$ is only about $\\sim5$ p$\\Omega$m\/T for $t_{\\text{Pt}}=10$ nm and $T=300$ K~\\cite{meyer2015anomalous}. This results in an AH signal $V_{\\text{AHE}}\\sim$ 0.3 $\\mu$V, considering a Hall bias of 2 mA and a magnetoelectric node voltage $V_{\\text{ME}}=0.3$ V. The Hall signal can be raised to $\\sim$ 1 $\\mu$V by increasing $V_{\\text{app}}$ to 1 V, and further enhanced by applying a larger Hall bias. However, doing so would negatively impact the energy consumed in the read operation. Sensing such a low $\\mu$V-range AH signal would require sophisticated instrumentation sense amplifiers that are area- and power-prohibitive\n(e.g., 2.5 mm$^2$ area and $\\sim$mW-range power~\\cite{witte2008current}).\n\nThis problem can be addressed by exploring other material systems with much higher interfacial spin-orbit coupling (SOC), resulting in larger AH coefficients. \nIn~\\cite{zhang2014effective}, a Pt\/Co\/Pt tri-layer is shown to exhibit $R_{\\text{s}}\\sim7.3\\times 10^{-10}$ $\\Omega$m\/T at 300 K for $t_{\\text{Co}}\\sim$ 10 nm, resulting in $V_{\\text{AHE}}\\sim$ 43.8 $\\mu$V at a Hall bias of 2 mA and $V_{\\text{ME}}=0.3$ V. \nMagnetic semiconductors like EuTiO$_3$ possess higher $R_{\\text{s}}\\sim$ $8\\times 10^{-9}$ $\\Omega$m\/T for $t_{\\text{EuTiO}_3}=$ 25 nm~\\cite{takahashi2018anomalous}. However, AH signals in such samples have been detected only at very low temperatures, of 2K, at which the ME effect in Cr$_2$O$_3$ vanishes.\n \nThe Hall signal could be improved in a topological insulators (TI) due to the presence of high SOC-enhanced surface states. \nFor example, the Bi$_2$Se$_3$\/LaCoO$_3$ stack considered in~\\cite{zhu2018proximity} demonstrates $R_{\\text{s}}$ as high as $\\sim1.59$ $\\mu\\Omega$m\/T at 100 K for $t_{\\text{Bi}_2\\text{Se}_3}\\sim$ 20 nm. \nThis results in a substantial improvement in the AH signal generated (i.e., $\\sim47.7$ mV). \nThe AH effect in the Bi$_2$Se$_3$\/LaCoO$_3$ interface is ascribed to the exchange coupling between the Bi$_2$Se$_3$ layer and the ferromagnetic LaCoO$_3$ layer via the proximity effect, and is enhanced by the high interfacial SOC. \nSimilarly, the (BiSb)$_2$Te$_3$\/TIG system considered in~\\cite{tang2017above} achieves a mV-range AH signal, though much closer to room temperature.\nA comparison of $R_{\\text{s}}\/t$ in various material systems is illustrated in Fig.~\\ref{fig:R_AHE}. \nAs can be inferred, TIs are an ideal material candidate to implement the AH read-out layer with Cr$_2$O$_3$ due to the potential of a $\\sim$mV-range AH signal, which can be easily read-out using a normal current latch sense amplifier~\\cite{kobayashi1993current}, i.e., without the need for sophisticated sensing equipment.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.33]{figures\/R_AHE_new.pdf}\n\\caption{Comparison of the AH coefficient per unit thickness and AH signal magnitude in different material systems. The AH signal $V_{\\text{AHE}}$ is calculated for a Hall bias of 2 mA and a magnetoelectric node voltage $V_{\\text{ME}}\\sim$ 0.3 V. TIs with high interfacial SOC exhibit greater AH coefficients and can generate large AH signals, capable of being detected by conventional current sense amplifiers.}\n\\label{fig:R_AHE}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.37]{figures\/MERAM_cell_comparison.pdf}\n\\caption{Benchmarking the ME-AFMRAM cell considered in this work against current state-of-the-art ME-AFMRAM technology, and trends in other emerging non-volatile storage devices from~\\cite{wong2016stanford}. Some important data points in this plot, representing the advances in various NVMs, include~\\cite{jan2012high,gajek2012spin,liu2010ultrafast} for STT-MRAM, \\cite{aratani2007novel,lin2010novel,vianello2012sb} for CBRAM, \\cite{zhao2014ultrathin,sekar2014technology,goux2014role} for RRAM, and~\\cite{matsui2006ta2o5,kim2010high,xiong2013self} for PCM, respectively.\nThe future potential of ME-AFMRAM lies in achieving ultra-fast, coherent rotation-based reversal (sub-100 ps write delay and fJ write energy) through a combination of dimensional scaling and strain-augmentation.}\n\\label{fig:cell_comparison}\n\\end{figure}\n\n\\subsubsection{Coherent rotation-based reversal}\nThe $\\sim$ns-range write latency of the ME-AFMRAM cell can be improved drastically if the chromia order can be switched through coherent rotation. In this case, the entire chromia sample undergoes reversal homogeneously, rather than following the incoherent DW propagation. For $\\mathcal{F}_{d} > 4\\mathcal{K}$, the order parameter switches via damping of gyromagnetic precessions~\\cite{parthasarathy2019dynamics}. However, if $\\mathcal{F}_{d} < 4\\mathcal{K}$, magnetization could switch due to thermal activation.\nHere, the switching time is exponentially dependent on the energy barrier of the sample.\nIn any case,\nit is thermal activation that leads to retention errors.\n\nTo realize coherent rotation in chromia, the applied ME pressure must exceed $4\\mathcal{K} = 2.92\\times 10^4$ J\/m$^3$. For a magnetic field of 0.5 T and $\\alpha_\\mathrm{ME} = 3.1$ ps\/m, the electric field required for coherent rotation is $1.18 \\times 10^{10}$ V\/m. Unfortunately, such a high electric field could lead to dielectric breakdown of chromia, given that the breakdown strength of chromia is $\\sim 2\\times 10^8$ V\/m~\\cite{sun2017local}.\nA potential solution to this challenge is to reduce the effective anisotropy of the sample such that the required threshold electric field scales down. This can be achieved through a variety of techniques, including substitutional alloying and the application of mechanical strain~\\cite{mu2019influence}. It is estimated that the write latency of a strain-augmented ME-AFMRAM cell can reach as low as a few 10's of ps. A comparison of the current state-of-the-art in ME-AFMRAM technology and its future potential versus trends in other emerging storage devices is presented in Fig.~\\ref{fig:cell_comparison}.\n\n\\subsubsection{Material and geometrical parameters of the chromia ME-AFMRAM cell}\n\nThe simulation parameters used in our SPICE models for the chromia ME-AFMRAM are listed in the following Table~\\ref{tab:params}.\n\n\\begin{table}[ht]\n\\centering\n\\footnotesize\n\\setlength{\\tabcolsep}{1mm}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{*{3}{c}}\n\\hline\n\\textbf{Parameter} & \\textbf{Value} & \\textbf{Ref.}  \\\\\n\\hline\nSaturation magnetization of Cr$_2$O$_3$, $M_{\\text{s}}$ & $2.6\\times 10^5$ A\/m & \\cite{artman1965magnetic} \\\\ \\hline\nMagnetoelectric coefficient of Cr$_2$O$_3$, $\\alpha_{\\text{ME}}$ & $3.1\\times 10^{-12}$ s\/m & \\cite{hehl2008relativistic}\\\\ \\hline\nUniaxial anisotropy energy of Cr$_2$O$_3$, $\\mathcal{K}$ & $7300$ J\/m$^3$  & \\cite{foner1963high} \\\\ \\hline\nGilbert damping constant of Cr$_2$O$_3$, $\\alpha_{\\text{G}}$ & $2\\times 10^{-4}$  & \\cite{belashchenko2016magnetoelectric}\\\\ \\hline\nThreshold ME pressure to depin DW, $\\mathcal{F}_{\\text{d}}$ & $25$ J\/m$^3$ & \\cite{parthasarathy2019dynamics} \\\\ \\hline\nApplied magnetic field, $H_{\\text{app}}$ & $0.5$ T & \\\\ \\hline\nApplied voltage, $V_{\\text{G}}$ & $0.3$ V & \\\\ \\hline\nLength of cell, $l$ & $60$ nm & \\\\ \\hline\nWidth of cell, $w$ & $60$ nm & \\\\ \\hline\nThickness of cell, $t$ & $10$ nm & \\\\ \\hline\nTemperature, $T$ & $292$ K & \\\\ \\hline\n$\\tau_\\mathrm{creep}$ (@ $\\mathcal{F} = 0$) & $\\sim1$ ms & \\\\ \\hline\n$\\tau_\\mathrm{flow}$ (@ $\\mathcal{F} = 74.2$ J\/m$^3$) & $\\sim0.22$ ns & \\\\ \\hline\n\\end{tabular}\n\\caption{Simulation parameters considered for the ME-AFMRAM cell.}\n\\label{tab:params}\n\\end{table}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.6]{figures\/AFMRAM_organization.pdf}\n\\caption{64KB ME-AFMRAM organization with 4$\\times$1 banks, 2$\\times$1 mats, 4$\\times$2 sub-arrays, and 128$\\times$64 bit cell arrays. Here, the word length is 128 bit. The memory organization is leveraged from~\\cite{dong2012nvsim}.}\n\\label{fig:MERAM_organization}\n\\end{figure*}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/AFMRAM_array.pdf}\n\\caption{Construction of the ME-AFMRAM cell array used in the memory architecture.\nThe signals BL$_{\\text{i,in}}$ serve to write data into the cells when Write Enable (WE) is on, and signals BL$_{\\text{i,out}}$ serve to read data from the cells when Read Enable (RE) is on.}\n\\label{fig:MERAM_array}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.43]{figures\/comparison_table.pdf}\n\\captionof{table}{Performance comparison of various memory technologies, from~\\cite{yang2013memristive, micron_nandflash, everspin_feram, chang2016resistance, kent2015new}. \nThe write and read latencies for ME-AFMRAM (DW model) are quoted for a 64KB memory with a 128-bit word line, simulated using NVSim~\\cite{dong2012nvsim}. \nThe energy-per-bit metric is for a single bit write onto a cell.}\n\\label{tab:Comparison_table}\n\\end{figure}\n\n\\subsection{ME-AFMRAM array}\n\nTo evaluate the system-level performance of ME-AFMRAM in the context of existing memory technologies, we simulate a 64KB DW-based ME-AFMRAM chip on NVSim, a standard tool for estimating the performance metrics of emerging NVMs~\\cite{dong2012nvsim}. \nThe organization of this 64KB memory, as leveraged from~\\cite{dong2012nvsim}, is shown in Fig.~\\ref{fig:MERAM_organization}. \nThe internal architecture of the ME-AFMRAM cell array, along with the peripheral decoders, drivers and sense amplifiers, constructed at the 15-nm CMOS node, is highlighted in Fig.~\\ref{fig:MERAM_array}. \nThe total write latency of the 64KB ME-AFMRAM, including the parasitics and peripheral latency (133.9 ps) and the dominant cell switching time ($\\sim$630 ps), is obtained as 763.9 ps from NVSim~\\cite{dong2012nvsim}. \nThe write latency can be improved by an order of magnitude via coherent rotation of the order parameter. \nThe total read latency of the chip, obtained from NVSim~\\cite{dong2012nvsim}, is $\\sim$2.3 ns. \nThis includes contributions from the sense amplifier (1.45 ns), bit-line parasitics (3.5 ps), decoders and other peripherals ($\\sim$150 ps), and the dominant AH measurement delay in the Bi$_2$Se$_3$ layer ($\\sim$0.7 ns)~\\cite{kikuchi2016anomalous}. \nState-of-the-art pulsed AH measurement schemes like~\\cite{kikuchi2016anomalous} are capable of operating in the GHz regime.\n\nThe output bit-line sensing can be achieved using a conventional current latch amplifier if a large-SOC material such as a TI is used to generate an AH signal in the range of tens of mV.\nThe read\/write endurance of the ME-AFMRAM is expected to be similar to that of STT-MRAM. A comparison of the performance metrics of the ME-AFMRAM with other memory technologies at the chip-level is presented in Table~\\ref{tab:Comparison_table}. It can be seen that the ME-AFMRAM offers some competitive advantages over other NVMs as well as over conventional memory systems.\n\n\n\\section{Application as Secure Memory}\n\\label{sec:security}\n\nAfter conducting cell- and array-level modeling and benchmarking of the chromia-based ME-AFMRAM, we continue with the implementation of the proposed SMART memory using the ME-AFMRAM.\n\n\\subsection{Threat model}\n\nFirst, we discuss the threat model, defining the strengths and capabilities of attackers, as well as the objectives and consequences of a successful attack. Most but not all attack scenarios presented here are\nspecific to NVMs.\n\n\\begin{itemize}\n\n\\item Attackers can launch cold-boot attacks~\\cite{halderman2009lest}.\nDuring power-down, there is some latency after the power-down sequence initiates until the moment when memory contents are completely secured. An attacker might use this gap to read out memory contents. To circumvent such attacks, memory encryption is typically employed~\\cite{chhabra2011nvmm,swami2018acme}.\n\n\\item Attackers could leverage properties like sensitivity to magnetic fields and temperature fluctuations to corrupt the data or induce a DoS~\\cite{jang2015self}. \nThey may forcibly write specific data patterns \nto memory, which accelerates aging and \ncauses memory failures.\n\n\\item With access to failure analysis equipment, attackers can also resort to advanced invasive attacks. \nThe majority of such attacks\ntarget at the back-end-of-line (BEOL), approaching from the top-most metal layer, which is also referred to as front-side attacks. \nVarious countermeasures have been proposed to protect the front-side, which include protective meshes, shields, and sensors~\\cite{lee19_shield,weiner18}.\nIn any case, \\textit{bus snooping} attacks are considered beyond the scope of this work.\n\n\\item Power-dissipation signatures when reading\/writing `0' and `1' within the NVM can be exploited for side-channel attacks to infer the data, through techniques like differential power analysis (DPA)~\\cite{kocher1999differential} and correlation power analysis (CPA)~\\cite{brier2004correlation}.\n\n\\end{itemize}\n\n\\subsection{Magnetic field and temperature attacks}\n\\label{sec:Mag}\n\nSTT-MRAMs have FM-based MTJs as their basic building blocks. FMs possess a macroscopic magnetization (or magnetic signature) that can be probed or inferred with using an external magnetic field. \nHence, magnetic fields can be used to infer or tamper with the stored data or even cause malfunctions in STT-MRAMs~\\cite{jang2015self}.\nStray magnetic fields as small as 10 mT could cause an unintended bit flip in STT-MRAM cells. Figure~\\ref{fig:STTMRAM} shows the magnetic field-induced bit flip in a representative FM, obtained by solving the Landau-Lifshitz-Gilbert equation for the FM dynamics~\\cite{ament2016solving}.\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[Trajectory for magnetic field-induced switching of a FM.]{%\n\\label{fig:STTMRAM_traj}%\n\\includegraphics[scale=0.18]{figures\/STTMRAM_traj.pdf}}%\n\\hspace{1ex}\n\\subfigure[Components for magnetic field-induced switching of a FM.]{%\n\\label{fig:STTMRAM_switching}%\n\\includegraphics[scale=0.23]{figures\/STTMRAM_switching.pdf}}%\n\\caption{The FMs in an STT-MRAM can be switched easily using external magnetic fields.}\n\\label{fig:STTMRAM}\n\\end{figure}\n\nAFMs, on the other hand, exhibit no external magnetic signature since their equal and opposite sublattice moments cancel each other out. \nHence, the bulk order parameter cannot be affected by external magnetic fields. \nTo switch the bulk order, staggered fields (opposite sign on opposite sublattices) must be applied on both the sublattice moments, as illustrated in Fig.~\\ref{fig:AFMRAM_field} inset. \nHowever, an external, homogeneous magnetic field is unable to provide such a staggered field arrangement, and hence, ends up canting the sublattice moments in a way wherein the torque due to the external field is exactly balanced by the exchange torque exerted by one sublattice moment on the other~\\cite{baltz2018antiferromagnetic}. \nSince external magnetic fields are unable to reorient the AFM order parameter, the SMART ME-AFMRAM is expected to be resistant to magnetic field attacks described in~\\cite{jang2015self}. \nWe note that switching the ME-AFM surface magnetization state using a combination of $E$ and $H$ fields would require an exact knowledge of the write cycles and the prior state of the surface, as well as\nmeans to control the electric field explicitly, which is to be concealed from an attacker. \n\nWith regards to temperature fluctuation-based attacks, an adversary might increase the ambient temperature of the ME-AFMRAM in an attempt to alter the stored data. \nNote that the N\\'{e}el temperature of pure chromia is 308 K~\\cite{shi2009magnetism}, above which the AFM ordering is destroyed. Hence, the attacker may corrupt the memory by heating it above the N\\'{e}el temperature. \nTo counter this, we consider Boron-doped chromia, whose N\\'{e}el temperature is demonstrated experimentally to be $\\sim400$ K~\\cite{street2014increasing}. \nHence, Boron-doped chromia can increase the resilience of SMART memory against temperature fluctuations. \nThat is because such larger temperature fluctuations (above 400 K) are easier to detect, and countermeasures like interception of such attacks become more feasible.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.3]{figures\/AFM_field_combined.pdf}\n\\caption{The application of a magnetic field is unable to switch the AFM order parameter, even when increasing the field magnitude. \nInset: (a) an external, homogeneous magnetic field may cant the sublattice moments, but it is incapable of rotating the AFM order; \n(b) staggering fields on the sublattice moments produce staggered tangential torques, which can reorient the AFM order.}\n\\label{fig:AFMRAM_field}\n\\end{figure}\n\n\\subsection{Data confidentiality attacks}\\label{Encrypt}\nAs with all NVMs, data remanence in the SMART memory could be exploited by attackers to steal sensitive information. The most effective countermeasure against such data confidentiality attacks, including cold-boot and stolen memory-modules attacks, is to encrypt the data using a secure encryption scheme before storing it in the memory. Advanced memory encryption techniques like counter mode encryption (CME) use block ciphers such as Advanced Encryption Standard (AES) to encrypt a seed using a secret key, in order to generate a one-time pad (OTP). \nThe seed for each write on a memory line consists of a secret key, the line address, and a counter value associated with that line, which is incremented with each subsequent write to the same line. Hence, the generated OTP is unique for each line address, and also for each write operation to the same address. \nThe OTP is then XOR-ed with the plaintext to obtain the ciphertext, which is stored in the non-volatile main memory.\nNote that the secret key used in the AES core is considered inaccessible to the attacker.\n\nDirectly applying XOR-based CME scheme to the SMART memory would result in large encryption overheads. This is because the CME scheme is tailored for NVMs like PCM and STT-MRAM, whose write time is on the order of $\\sim$ns. The access latency of ME-AFMRAM is sub-ns for DW-based propagation and few 10's of ps for coherent rotation. A general encryption scheme for SMART memory, switching either via DW propagation or coherent rotation, must be such that the overall memory access latency is not adversely affected. Existing encryption solutions based on CMOS XOR gates with 10's of ps delay are rendered ineffective as their encryption time is comparable to the memory write time, resulting in idle clock cycles.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.3]{figures\/Memcryption_scheme.pdf}\n\\caption{(a) CME uses AES to generate an OTP, using the memory line address, a counter, and a secret key. The encryption and decryption is performed outside the non-volatile main memory (NVMM).\n(b) \\textit{Memcryption} uses a secret key and the line address as seed for AES, to generate an encryption pulse. \nThat pulse is used to control the bitwise operation of CNOT gates, and is embedded in the data path within the NVMM.}\n\\label{fig:Memcryption_scheme}\n\\end{figure}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.42]{figures\/Memcryption.pdf}\n\\caption{SMART memory architecture with \\textit{Memcryption}.\nThe CNOT layer for decryption is not shown for simplicity.}\n\\label{fig:Memcryption}\n\\end{figure*}\n\nHere, we propose to use in-memory encryption, or \\textit{Memcryption}, using bitwise CNOT (i.e., controlled-NOT) gates constructed from ME-AFM-based logic. \nBy tying the encryption pulse to the control signals of CNOT gates, one can achieve such \\textit{Memcryption}.\nSpin devices like the ME-AFM transistor~\\cite{dowben2018towards} are able to implement polymorphic logic gates, which can provide inverting or non-inverting functionality based on a control signal~\\cite{patnaik2018advancing,patnaik2019spin}. \nHence, the ME-AFM transistor is used to realize the CNOT gate. Further, the ME-AFM transistor is shown to exhibit delays as small as $\\sim10$ ps, which is substantially faster than CMOS XOR gates and compatible with the SMART memory write-times.\nSuch homogeneity in the technology and materials by using only ME-AFM for both the memory cells and the CNOT gates will ease the fabrication. \nIn \\textit{Memcryption}, we embed ME-AFM transistor-based CNOT gates directly in the data path connected to the memory array; hence, the encryption is in-memory, as opposed to prior works using a separate encryption block. \nThis integration of encryption and memory array is not detrimental to the memory density since ME-AFM transistors have a footprint that is substantially smaller than that of CMOS XOR gates.\nFigure~\\ref{fig:Memcryption_scheme} contrasts our \\textit{Memcryption} scheme with prior CME techniques. \n\nThe SMART memory architecture with \\textit{Memcryption} is shown in Fig.~\\ref{fig:Memcryption}.\nA trusted 128-bit key, provided and stored within a secure processing module (SPM) along with the processor, is concatenated with the memory address and used as seed for AES.\nThe AES core, which is to be integrated on the NVM chip,\\footnote{Heterogeneous spin-CMOS integration is not prohibitive since the underlying AFM technology is compatible with \nCMOS processes in the BEOL. \nIn general, hybrid spin-CMOS designs have been explored in prior works~\\cite{yogendra2015domain}.} thus produces an encryption pulse whose bits are used as the control bits for the CNOT gates of the in-memory encryption layer.\nDepending on the control bits, the encryption layer flips bits selectively in the plaintext before performing a memory-write. \nDuring decryption, the same encryption pulse is generated again and used to perform bitwise CNOT operations on the ciphertext (read from memory), to obtain the plaintext.\n\nA comparison of the \\textit{Memcryption} scheme versus CME (when also applied to ME-AFMRAM) is presented in Table~\\ref{overhead_comparison}. \nThe array considered is a 128-bit ME-AFMRAM, while the AES and CMOS peripherals are synthesized using the 15nm \\textit{NanGate} technology.\nWe observe that Memcryption with SMART memory \nhas a better encryption latency than CME, which utilizes regular CMOS XORs.\nWe also note that \\textit{Memcryption} helps reduce the encryption latency but is similar to CME with respect to the energy overheads. \nThat is because energy dissipation is dominated by the AES core in any case.\nWe also reiterate that \\textit{Memcryption} is tailored specifically as a memory-side scheme \nfor ME-AFMRAM, to achieve low encryption latency, owing to the homogeneous delays of the memory array and the encryption layer. \nHowever, it may not serve well as an efficient implementation for any generic NVM.\n\nWith regards to the reliability and lifetime of the ME-AFMRAM used to construct the SMART memory, its endurance is comparable to that of STT-MRAM. \nHowever, it also suffers from the same errors that plague the STT-MRAM, i.e., faults in the peripheral CMOS circuitry including the access transistors~\\cite{chintaluri2016analysis}. \nTo address these faults and ensure the correctness of the stored data, standard error correction techniques for NVMs~\\cite{swami2017reliable} like the error correction pointer (ECP) and other advanced schemes based on ECP, including ``Pay-As-You-Go''~\\cite{qureshi2011pay} and ``Zombie memory''~\\cite{azevedo2013zombie}, can be implemented memory-side and integrated on the ME-AFMRAM array. \nThe ECP memory can be realized using homogeneous spintronics technology, including the STT-MRAM or the ME-AFMRAM itself, or by leveraging heterogeneous spin-CMOS integration.\n\n\\begin{table}[ht]\n\\centering\n\\footnotesize\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{*{3}{c}}\n\\hline\n\\textbf{Encryption technique}\n& \\textbf{Latency} & \\textbf{Energy} \\\\\n\\hline\nCME~\\cite{chhabra2009making} &  299.23 ps (2.99$\\times$) & 17.371 pJ \\\\ \\hline\nMemcryption &  273.46 ps (2.73$\\times$) & 17.370 pJ \\\\ \\hline\n\\end{tabular}\n\\caption{Comparison for latency and energy when applying the CME and \\textit{Memcryption} schemes to a 128-bit ME-AFMRAM array. The baseline latency for the unencrypted array is $\\sim 100$ ps.}\n\\label{overhead_comparison}\n\\end{table}\n\n\\subsection{Power side-channel attacks}\n\\label{Power}\n\nAsymmetric read\/write characteristics in NVMs like STT-MRAM make them susceptible to side-channel attacks which exploit the different signatures incurred when reading\/writing `1's \nand `0's bits. STT-MRAMs employ MTJs with a fixed FM reference layer, \nwith another free layer either oriented parallel or anti-parallel to that reference layer. Depending on the relative orientation of these two layers,\nthe MTJ falls into a low or high resistance state; the low or high state corresponds to logic `0' or logic `1' state, respectively. Hence, the currents drawn for read\/write operations are different depending on reading\/writing a `0' or a `1'. \nThus, an attacker could attach a resistor in a voltage-divider configuration with\nthe MTJ cell, monitor the voltage drops across that resistor, and perform DPA to recover the data being written to or read from the cell. In fact, such an attack was showcased against an STT-MRAM-based cache in~\\cite{khan2017side}.\n\nFor the SMART memory, recall that writing is achieved using electrical fields, not currents. Further, the electric-field magnitude required for writing `0's and `1's is equivalent; see write voltage and polarization voltage traces in Fig.~\\ref{fig:AFMRAM_timing}. This is because there is no reference layer or tunneling magnetoresistance in the ME-AFMRAM, which would cause asymmetricity. As for the read operation, the proximity effect-induced moment in the Pt electrode is slightly different for reading `0' or `1'. \nHowever, this imbalance in the Hall signals can be compensated for by introducing appropriate offsets in the Hall measurement setup, as demonstrated in~\\cite{kosub2017purely}. Hence, the SMART memory can achieve symmetric signatures for both read and write and for both `0$\\rightarrow$1' and `1$\\rightarrow$0' \ntransitions, thus thwarting any DPA-based power side-channel attacks.\n\n\\subsection{Photonic side-channel and \nbackside attacks}\n\\label{Photonic}\n\nLeveraging the photonic side-channel (PSC) to circumvent the security guarantees provided by cryptographic algorithms like AES and RSA has been demonstrated recently~\\cite{ferrigno2008aes,schlosser2013simple}. Simple Photonic Emission Analysis (SPEA) or Differential Photonic Emission Analysis (DPEA) can be carried out using photo-emission equipment available for similar cost as that of power-analysis equipment. The essence of the PSC is to observe photo-emissions emanating for switching of CMOS transistors.\nFor SRAM- or DRAM-based memories, this emission can then be correlated with the data being programmed into the memory. In~\\cite{ferrigno2008aes}, the PSC was found to originate when kinetic energy gained by charge carriers in the transistor channel is transferred to photons, which are visible through photo-detectors. In~\\cite{schlosser2013simple}, the authors leveraged this \ninformation to perform a side-channel attack, ultimately recovering the full AES key. Modern-day chips use several metal layers, which interfere with the emission of photons from the frontside of any integrated circuit (IC); therefore, a natural direction is to observe the photon emission from the backside of ICs. \n\nWhile CMOS-based memory technologies like SRAM and DRAM are prone to such PSC attacks, the SMART memory is AFM-based and involves no photonic emissions emanating from transistor channels. Data read-out in the SMART memory can only be accomplished through an AH measurement setup. Further, even if an advanced attacker is able to isolate the SMART memory cell and gain access to the AH setup from the frontside, they would only be able to recover the encrypted ciphertext (as described in Sec.~\\ref{Encrypt}).\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we present \\textit{SMART: A Secure Magnetoelectric Antiferromagnet $\\!$-Based Tamper-Proof Non-Volatile Memory}, by utilizing the unique properties of ME-AFMs. \nThe ME-AFMRAM, which is at the core of the SMART memory, has an access latency of sub-1 ns (for DW-based switching) down to only 10's of ps (for coherent rotation switching) with an energy-per-bit of $\\sim$ 0.13 pJ. \nBesides its superior performance as compared to prior NVMs like STT-MRAM and PCM, the SMART memory exhibits no sensitivity to external magnetic fields, which makes it resilient to magnetic field-based data tampering and denial of memory service attacks that commonly plague other ferromagnets-based NVMs. \nTo solve the security vulnerability of data remanence (after power-down) in the SMART memory, we demonstrate a new encryption technique called \\textit{Memcryption}. \nThis scheme employs emerging ME-AFM-based logic to implement a CNOT-centric in-memory encryption, which is particularly tailored to reduce the encryption and decryption latency in the SMART memory. \nFurther, symmetric read and write signatures for `0' and `1' bits render prominent side-channel attacks like the differential power attack futile against the SMART memory. \nAdvanced photonic side-channel attacks, which are powerful threats against any CMOS IC by observing all internal transistor activity from the frontside or backside, are ineffective against the SMART memory due to the fundamentally different switching mechanism as well as the proposed \\textit{Memcryption} safeguard.\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nLet $C$ be a smooth complex curve of genus $g>1$,\ndenote by $C^2$ the cartesian product of\n$C$ with itself, by $C^{(2)}$ the \nsecond symmetric product of $C$  \nand by $\\pi\\colon C^2\\to C^{(2)}$ the\ndouble cover $(p,q)\\mapsto p+q$.\nLet $\\Delta$ be the image\nof the diagonal via $\\pi$ and  $K$ be\na canonical divisor of $C^{(2)}$.\nIn this paper we are interested\nin the finite generation of the {\\em extended\ncanonical ring}:\n\\[\n R(\\Delta,K)\n \\, :=\\,\n \\bigoplus_{(a,b)\\in{\\mathbb Z}^2}H^0(C^{(2)},a\\Delta+bK).\n\\]\nIt is not difficult to show that finite generation\nis equivalent to ask that the two-dimensional \ncone $\\Nef(\\Delta,K)$ consisting of classes \nof nef divisors within the rational vector space \nspanned by the classes of $\\Delta$ and $K$ \nmust be generated by two semiample classes\n(see proof of Theorem~\\ref{teo-1}).\nThe Kouvidakis conjecture~\\cite{CK} states that,\nfor the very general curve, if $\\Nef(\\Delta,K)$ is \nclosed then $g$ must be a square, \ni.e. $g = (k-1)^2$ for $k>1$ integer. \nWe thus focus our attention to\nthe case of square genus and assume $C$\nto be a complete intersection of a \nquadric $Q$ with a degree $k>2$ hypersurface.\nWe denote by $\\eta_C$ the class of the\ndifference of the two $g_k^1$ of $C$\ninduced by the two rulings of $Q$, which\nis trivial if $Q$ is a cone.\nOur first result is the following.\n\\begin{introthm}\n\\label{teo-1}\nLet $C\\subseteq{\\mathbb P}^3$ be a smooth complete \nintersection of a quadric with a degree $k>2$\nsurface. Then the following are equivalent.\n\\begin{enumerate}\n\\item\n$R(\\Delta,K)$ is finitely generated.\n\\item\n$\\eta_C$ is torsion non-trivial.\n\\end{enumerate}\nMoreover if both the above conditions are satisfied\nthen $\\eta_C$ has order at least $k$.\n\\end{introthm}\nLet $\\mathcal F_k$ be the  open subset \nof the Hilbert scheme of curves of bi-degree $(k,k)$ \nof ${\\mathbb P}^1\\times{\\mathbb P}^1$ consisting of smooth curves\nand let $\\mathcal F_k^{\\rm tor}\\subseteq\\mathcal F_k$\nbe the subset consisting of curves $C$ such that\nthe class $\\eta_C$ of the difference between the\ntwo $g_k^1$ is torsion.\nOur second theorem is the following.\n\\begin{introthm}\n\\label{teo-2}\nThe locus $\\mathcal F_k^{\\rm tor}$ is \na countable union of subvarieties of complex dimension \n$\\geq 4k-1$ and the set of subvarieties of\ndimension $4k-1$ is dense in $\\mathcal F_k$\nin the analytic topology.\n\\end{introthm}\nThe paper is organized as follows.\nIn Section~\\ref{sym}, after recalling some basic facts\nabout the symmetric product of a curve,\nwe prove Theorem~\\ref{teo-1}. In Section~\\ref{grid-fam}\nwe introduce the grid family consisting of \ncurves of bi-degree $(k,k)$ on a smooth quadric\nwhich pass through a complete intersection\nof type $(k,0),(0,k)$. We show that  the\ngrid family is exactly the subvariety  \nof  $\\mathcal F_k^{\\rm tor}$ corresponding to \ntorsion of order $k$ and \nhas the expected dimension $4k-1$.\nSection~\\ref{density} is devoted to the proof of Theorem~\n\\ref{teo-2}. In Section~\\ref{hyp} we prove a density\ntheorem for hyperelliptic curves, providing\nan alternative proof for Theorem~\\ref{teo-2}\nin case $g=4$ (see Corollary~\\ref{cor}). \nThis result has an independent \ninterest and is proved in the spirit of Griffiths \ncomputations of the infinitesimal invariant~\\cite{Gr}.\nFinally, in Section~\\ref{exa} \nwe consider examples\nof curves $C$ with $\\eta_C$ torsion.\n\nIn all the paper we work over the field of complex \nnumbers except for Section~\\ref{sym} (see Remark~\\ref{ch}).\n\n\n\\section{The second symmetric product}\n\\label{sym}\nLet $C$ be a smooth projective \ncurve of genus $g>1$ defined over an algebraically \nclosed field ${\\mathbb K}$ of characteristic $0$.\n\\begin{proposition}\\label{pic}\nThe diagonal embedding \n$\\imath\\colon C\\to C^{(2)}$\ninduces an isomorphism \n$\\imath^*\\colon \\Pic^0(C^{(2)})\\to\n\\Pic^0(C)$ of abelian varieties.\n\\end{proposition}\n\\begin{proof}\nTo prove the statement we explicitly construct\nthe inverse map of $\\imath^*$. Given a point\n$p\\in C$ let $H_p$ be the curve of $C^{(2)}$\nwhich is the image of $\\{p\\}\\times C$ via $\\pi$.\nDefine the map ${\\rm Div}(C)\\to {\\rm Div}(C^{(2)})$ by\n$\\sum_in_ip_i\\mapsto\\sum_i n_iH_{p_i}$ and\nobserve that it maps principal divisors to principal \ndivisors. The induced map of Picard groups  \nrestricts to a homomorphism $\\Pic^0(C)\\to\n\\Pic^0(C^{(2)})$ which is easily seen to be a \nright inverse of $\\imath^*$.\nSince the two abelian varieties\n$\\Pic^0(C)$ and $\\Pic^0(C^{(2)})$ have \nthe same dimension we conclude that\n$\\imath^*$ is an isomorphism.\n\\end{proof}\nObserve that $\\Delta$ is the branch divisor\nof the double cover $\\pi$ and thus \nits class is divisible by $2$ in $\\Pic(C^{(2)})$.\nMoreover the following linear equivalences\n\\[\n \\Delta|_\\Delta\n \\sim\n -2K_\\Delta\n \\qquad\n \\qquad\n K|_\\Delta\n \\sim\n 3K_\\Delta\n\\]\ncan be proved by passing to $C^2$ \nand calculating the restriction of $K_{C^2}$\nto the diagonal.\nBy the Riemann-Hurwitz formula we get the equalities\n$2(2g-2)^2 = K_{C^2}^2 = 2 (K + \\frac{\\Delta}{2})^2$ \nfrom which we deduce the following \n\\begin{equation}\n \\label{intersections}\n K^2 = (g-1)(4g-9)\n \\qquad\n K\\cdot\\Delta = 6(g-1)\n \\qquad\n \\Delta^2 = -4(g-1).\n\\end{equation}\nIn particular the classes of $\\Delta$ and $K$ are \nindependent in the N\\'eron-Severi group of ${C^{(2)}}$.\nWe let $\\langle \\Delta, K\\rangle$ be the rational \nvector subspace of $\\Pic({C^{(2)}})\\otimes_{\\mathbb Z}{\\mathbb Q}$\ngenerated by the classes of $\\Delta$ and $K$\nand form the following cone\n\\[\n {\\rm Nef}(\\Delta,K)\n =\n \\{D\\in\\langle \\Delta, K\\rangle\\, :\\, D\\text{ is nef}\\}.\n\\]\nThis cone is related to the Kouvidakis conjecture\nwhich predicts which ones are the extremal rays of \n${\\rm Nef}(\\Delta,K)$. In case the genus is a square,\ni.e. $g = (k-1)^2$, the conjecture is known to be\ntrue~\\cite{CK} for a very general curve $C$ and it \nholds as well if $C$ has an irreducible $g_k^1$,\nthat is the curve~\\eqref{eq:gamma} defined below\nis irreducible.\nIn this case the extremal rays of ${\\rm Nef}(\\Delta,K)$ \nare spanned by the classes of $2K+3\\Delta$ and \n$2K+(5-2k)\\Delta$.\n\n\\begin{proposition}\n\\label{ray-1}\nLet $C$ be a smooth curve of genus at least\ntwo. Then the divisor \n$2K+3\\Delta$ of $C^{(2)}$ is semiample.\n\\end{proposition}\n\\begin{proof}\nObserve that the divisor $2K+3\\Delta$\nis big since $K$ is ample and $\\Delta$\nis effective. Moreover it is nef since \n$(2K+3\\Delta)\\cdot\\Delta = 0$.\nWe have an exact sequence of sheaves\n\\[\n \\xymatrix@1{\n  0\\ar[r]\n  &\n  {\\mathcal O}_{C^{(2)}}(2K+2\\Delta)\\ar[r]\n  &\n  {\\mathcal O}_{C^{(2)}}(2K+3\\Delta)\\ar[r]\n  &\n  {\\mathcal O}_\\Delta\\ar[r]\n  &\n  0.\n }\n\\]\nSince $2K+2\\Delta = N + K+\\frac{1}{2}\\Delta$,\nwith $N=\\frac{1}{2}(2K+3\\Delta)$ nef and big,\nthen by the Kawamata-Viehweg vanishing\ntheorem and the long exact sequence in \ncohomology of the above sequence we \nconclude that $\\Delta$ is not contained \nin the base locus of $|2K+3\\Delta|$.\nThe statement follows by the ampleness \nof $K$ and the Zariski-Fujita theorem\n~\\cite[Remark 2.1.32]{La}.\n\\end{proof}\n\n\nAssume now that $C$ is a smooth curve\nof genus $g = (k-1)^2>1$ which admits a \n$g_k^1$ and define the following curve \nof $C^{(2)}$:\n\\begin{equation}\n\\label{eq:gamma}\n \\Gamma \\, :=\\, \\{p+q : g_k^1-p-q\\geq 0\\}.\n\\end{equation}\nIt can be easily proved that $\\Gamma$ is irreducible \nif the $g_k^1$ does not contain a $g_r^1$ with $r<k$.\nIn particular this holds if the $g_k^1$ is simple.\nObserve that the $g_k^1$ defines a morphism\n$\\Gamma\\to{\\mathbb P}^1$ of degree $\\frac{1}{2}k(k-1)$\nwhose branch points are exactly those\nof the $g_k^1$. Thus if the $g_k^1$ is simple\nwe deduce that\n\\begin{equation}\n\\label{genus}\n 2g(\\Gamma)-2 = -k(k-1)+2(k-2)(k-1)k,\n\\end{equation}\nwhere $g(\\Gamma)$ is the genus of $\\Gamma$.\nBy continuity the genus of $\\Gamma$ stays the \nsame even if the $g_k^1$ is not simple.\n\n\\begin{lemma}\\label{gamma}\nIf $C$ admits a $g_k^1$ then the divisor\n$2K+(5-2k)\\Delta$ is numerically equivalent to \n$(4k-8)\\Gamma$.\nMoreover if $C\\in\\mathcal F_k$, the curves  \n$\\Gamma$ and $\\Gamma'$ corresponding to the\ntwo distinct $g_k^1$ on $C$ are disjoint.\n\\end{lemma}\n\\begin{proof}\nLet $H$ be the curve of $C^{(2)}$ defined by\n$\\{p+q : q\\in C\\}$. By the Riemann-Hurwitz formula one\nhas the numerical equivalence\n\\[\n K\n \\equiv\n (2g-2)H-\\frac{1}{2}\\Delta.\n\\]\nThus by the genus formula $2g(\\Gamma)-2\n= \\Gamma^2+(2g-2)\\Gamma\\cdot H-\\frac{1}{2}\n\\Delta\\cdot\\Gamma$, by Equation~\\eqref{genus}\nand the equalities $\\Gamma\\cdot H = k-1$,\n$\\Delta\\cdot\\Gamma = 2g-2+2k$ we deduce\n$\\Gamma^2=0$. Since the intersection matrix\nof the divisors $\\Gamma,K,\\Delta$\n\\[\n \\left(\n  \\begin{array}{rrr}\n   0 & (2k-5)(k-1)k & 2(k-1)k\\\\\n   (2k-5)(k-1)k & (2k-5)(k-2)k(2k+1) & 6(k-2)k\\\\\n   2(k-1)k & 6(k-2)k & -4(k-2)k\n  \\end{array}\n \\right)\n\\]\nhas rank two, by the Hodge Index theorem $\\Gamma$ is\nnumerically equivalent to a rational linear\ncombination of $K$ and $\\Delta$.\nMoreover, being the kernel of the above matrix\ngenerated by the vector $(4k - 8, -2, 2k - 5)$, we\nget the first statement. \nObserve that  $\\Gamma$ and $\\Gamma'$  \nhave no common component since \notherwise the map from $C$ to the quadric \ngiven by the two $g_k^1$ \nwould not be an embedding.\nThus the second statement \nimmediately follows from the equality\n$\\Gamma\\cdot\\Gamma' = \\Gamma^2=0$.\n\\end{proof}\n\n \n \n\\begin{lemma}\\label{fib}\nIf $f\\colon C^{(2)}\\to Y$ is a fibration with connected \nfibers onto a smooth curve, then $Y$ has genus \nat most one. If moreover a fiber of $f$ is numerically \nequivalent to a rational linear combination of \n$\\Delta$ and $K$ then $Y$ is rational.\n\\end{lemma}\n\\begin{proof}\nTo prove the first statement observe that if\n$Y$ had genus $\\geq 2$, then there would\nbe two linearly independent holomorphic \n$1$-forms $w_1,w_2$ of $C^{(2)}$,   \nobtained by pull-back of $1$-forms of $Y$, \nsuch that $w_1\\wedge w_2 = 0$. On the other hand \nwe have a commutative diagram of isomorphisms\n\\[\n \\xymatrix{\n  {\\bigwedge^2 H^0(C^{(2)},\\Omega^1)}\n  \\ar[rr]^-{w_1\\wedge w_2\\mapsto w_1\\wedge w_2}\n  \\ar[rd]_\\alpha &&\n  H^0(C^{(2)},\\Omega^2)\\\\\n  & {\\bigwedge^2 H^0(C,\\Omega^1)}\\ar[ru]_\\beta \n }\n\\]\nwhere $\\alpha$ is induced by pull-back via the \ndiagonal embedding $C\\to C^{(2)}$ and $\\beta$\nis defined by $w_1\\wedge w_2\\mapsto \n\\frac{1}{4}(\\pi_1^*w_1\\wedge\\pi_2^*w_2+\n\\pi_2^*w_1\\wedge\\pi_1^*w_2)$, with $\\pi_i:C^2\\to C$ \nthe projections onto the two factors. \nThus such $1$-forms cannot exist and $Y$\nmust have genus at most one.\n\nTo prove the second statement, assume\nthat a fiber $F$ of $f$ is numerically equivalent \nto a rational linear combination of \n$\\Delta$ and $K$.\nIf $Y$ has genus one then we have the \nfollowing commutative diagram\n\\[\n \\xymatrix{\n C\\ar[r]^-\\imath\\ar[d]^-{\\alb_C} & C^{(2)}\\ar[r]^-f\\ar[d]^-{\\alb_{C^{(2)}}} & Y\\ar[d]^-{\\alb_Y}_-\\cong\\\\\n \\Alb(C)\\ar[r]^-{\\alb_\\imath}_-\\cong & \\Alb(C^{(2)})\\ar[r]^-{\\alb_f} & \\Alb(Y)\\\\ \n }\n\\]\nwhere $\\alb_\\imath$ and $\\alb_f$\nare induced by the universal property of \nthe Albanese morphism and $\\alb_\\imath$\nis an isomorphism by Proposition~\\ref{pic}.\nLet $\\Theta$ be the theta divisor of $\\Alb(C^{(2)})$.\nRecall that both $\\alb_{C^{(2)}}(\\Delta)$ and \n$\\alb_{C^{(2)}}(K)$ are numerically equivalent to\na multiple of $\\Theta^{g-1}$.\nBy our assumption on $F$ we deduce that\n$\\alb_{C^{(2)}}(F)\\equiv c\\,\\Theta^{g-1}$. Moreover\n$c$ is non-zero since $F$ cannot be \ncontracted by $\\alb_{C^{(2)}}$.\nBut this gives a contradiction, since \n$\\alb_f\\circ\\alb_{C^{(2)}}$ contracts $F$, while\n$\\alb_f(\\Theta^{g-1})$ cannot be a point, being the \nimage of $\\alpha_Y\\circ f\\circ\\imath$.\n\\end{proof}\n\nFor a proof of the following lemma see also\n~\\cite{BHPV}*{Lemma III.8.3}.\n\n\\begin{lemma}\n\\label{covering}\nLet $f\\colon S\\to\\mathbb {\\mathbb C}$ be a proper \nmorphism and let $nF$ be a multiple fiber \nof $f$ with multiplicity $n>1$. \nThen ${\\mathcal O}_F(F)$ is torsion non-trivial.\n\\end{lemma}\n\n\\begin{proof}\nWe first show that $\\mathcal L={\\mathcal O}_S(F)$ is not trivial.\nConsider the closure of the graph of $f$ \nin $S\\times \\mathbb P^1$, let \n$\\bar S$ be its minimal resolution and \n$\\bar f:\\bar S\\to \\mathbb P^1$ \nbe the fibration given by the projection \nto the second factor. \nAssume by contradiction that $F\\sim 0$ \nin $S$. Thus $F\\sim \\alpha F'$ in $\\bar S$,\nwhere $F'$ is a divisor with support contained \nin $\\bar S-S=\\bar f^{-1}(\\infty)$ and $\\alpha$ \nis a positive integer. \nSince $h^0(\\bar S, nF)=2$ with $n>1$, then  \n$h^0(\\bar S, F)=1$, giving a contradiction.\n\n\nThe line bundle $\\mathcal L$ thus defines a\nnon-trivial \\'etale cyclic covering \n$\\eta\\colon S'\\to S$.\nBy taking the Stein factorization of $f\\circ\\eta$\nwe get a commutative diagram\n\\[\n \\xymatrix{\n  S'\\ar[r]^-\\eta\\ar[d]^-{f'} & S\\ar[d]^-f\\\\\n  B\\ar[r]^-\\nu & {\\mathbb C},\n }\n\\]\nwhere $f'$ is a morphism with connected \nfibers and $\\nu$ is a finite map.\nIf $\\mathcal L|_{F}={\\mathcal O}_F(F)$ is trivial, then \n$\\nu$ is an\n\\'etale covering of $\\mathbb {\\mathbb C}$, \nsince the restriction of $\\mathcal L$ \nto any fiber of $f$ is trivial.\nThus  $\\nu$ is the trivial covering and $B$ \nhas $n$ connected components,\na contradiction since $\\eta$ is non-trivial.\n\nSince $\\mathcal L^{\\otimes n}$ is trivial,\nthen clearly its restriction to $F$ is trivial.\nThis concludes the proof. \n\\end{proof}\n\n\n\\begin{lemma}\\label{triv}\nLet $C$ be a non-hyperelliptic \ncurve of genus $g = (k-1)^2>1$ \nwhich is complete intersection \nof a quadric cone with a degree $k$\nsurface of ${\\mathbb P}^3$. Then the divisor\n$\\Gamma$ of $C^{(2)}$, \ncorresponding to the $g_k^1$ \nof $C$ defined by the ruling of \nthe cone, is not semiample.\n\\end{lemma}\n\\begin{proof}\nWe first show that the line bundle\n${\\mathcal O}_\\Gamma(\\Gamma)$ is trivial.\nIndeed let $Q_t\\subseteq{\\mathbb P}^3\\times\n\\mathbb A^1$ be a family of quadrics\nwhose central fiber $Q_0$ is the cone\ncontaining $C$ and whose general fiber \nis a smooth quadric. Let $\\mathcal D$\nbe a divisor of ${\\mathbb P}^3\\times\\mathbb A^1$\nwhich cuts out on the general fiber $Q_t$\na smooth curve $C_t$ of type $(k,k)$ with two\nsimple $g_k^1$ and $C$ on $Q_0$.\nThe family $\\mathcal C\\to\\mathbb A^1$\nof curves $C_t$ gives a family \n$\\mathcal C^{(2)}\\to\\mathbb A^1$\nwhose general fiber is $C_t^{(2)}$.\nOn any such fiber there are two \ncurves $\\Gamma_t$, $\\Gamma_t'$ \ncorresponding to the two $g_k^1$\non $C_t$. The line bundle \n${\\mathcal O}_{\\Gamma_t}(\\Gamma_t')$ is\ntrivial, by Lemma~\\ref{gamma},\nand its limit is ${\\mathcal O}_\\Gamma(\\Gamma)$,\nwhich proves the claim.\n\nAssume now, by contradiction, that \n$\\Gamma$ is semiample. \nSince $\\Gamma^2=0$,\na multiple $n\\Gamma$ defines a morphism\n$f\\colon C^{(2)}\\to B$, where $B$ is a curve. Moreover,\nafter possibly normalizing, we can assume $B$ \nto be smooth. Now, let $f=\\nu\\circ\\varphi$ be \nthe Stein factorization of $f$, where \n$\\varphi\\colon C^{(2)}\\to Y$ is a morphism with\nconnected fibers. By Lemma~\\ref{covering}\nand the fact that ${\\mathcal O}_\\Gamma(\\Gamma)$ is\ntrivial we deduce that $\\Gamma$ is a union \nof fibers of $\\varphi$. Moreover \nboth the hypotheses of Lemma~\\ref{fib} are satisfied, \nthus $Y$ must be a smooth rational curve.\nLet $H$ be the curve of\n$C^{(2)}$ which is the image of the curve \n$\\{p\\}\\times C$ via $\\pi$.\nThe equality $\\Gamma\\cdot H = k-1$ implies\nthat $\\varphi|_H$ is a covering of $Y$ whose degree \n$d$ divides $k-1$. \nThus $C\\cong H$ would admit two maps to \n${\\mathbb P}^1$ of degrees $k$ and $d$, respectively.\nBeing the degrees coprime, the curve $C$ \nwould be birational to a curve of bi-degree $(k,d)$\nof ${\\mathbb P}^1\\times{\\mathbb P}^1$, whose genus is smaller\nthan $g$, a contradiction.\n\\end{proof}\n\n \\begin{remark}\\label{rem1}\n If $D$ is a prime divisor on a projective \n surface $X$ such that $|D|$\n has dimension $0$ then $D$ is semiample \n if and only if $\\dim |nD| > 0$ for some $n>1$.\n Indeed the ``only if'' part is obvious, while \n the other implication follows from the fact\n that the fixed divisor of $|nD|$ is $mD$ \n for some $m<n$ and thus the base locus of\n $|(n-m)D|$ is at most zero-dimensional and\n one concludes by Zariski-Fujita theorem\n ~\\cite[Remark 2.1.32]{La} \n \\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{teo-1}]\nWe show that $(i)\\Rightarrow (ii)$ holds.\nAssume that the extended canonical ring\n$R := R(\\Delta,K)$ is finitely generated.\nWe begin to show that any divisor \nwhose class is in $\\Nef(\\Delta,K)$ \nis semiample. By Proposition~\\ref{ray-1}\nit suffices to show that $2K+(5-2k)\\Delta$\nis semiample, or equivalently that the divisor\n$\\Gamma$ defined by a $g_k^1$ of $C$\nis semiample, by Lemma~\\ref{gamma}.\nLet $f_1,\\dots,f_r$ be a minimal system of \nhomogeneous generators of $R$\nwith respect to the ${\\mathbb Z}^2$-grading\nand let $w_i := \\deg(f_i)\\in{\\mathbb Z}^2$ be\nthe degree of $f_i$ for any $i$.\nLet $f_1\\in R$ be a generator whose \ndegree belongs to the ray generated by \n$[\\Gamma]$ and let $D$ be a\nvery ample divisor of the form $a\\Gamma+b K$, with $a,b>0$\nwhose class $w$ lies in the interior of the cone  \n\n\n\\begin{minipage}{0.5\\textwidth}\n\\[\n \\bigcap_{i=2}^r{\\rm cone}(w_1,w_i)\n \\cap {\\rm cone}(w_1,[K])\n \\]\n\\end{minipage\n\\begin{minipage}{0.5\\textwidth}\n\\begin{center}\n \\begin{tikzpicture}[scale=0.65]\n  \\draw[-,thick] (0,0) -- (2,0) node[below]{$w_r$};\n  \\draw[-,thick] (0,0) -- (-2.2,1.1) node[left] {$w_1$};\n  \\draw[-,color=blue] (0,0) -- (0,1.8); \n  \\node[right] at (-0.6,2.2){$[K]$};\n  \\foreach \\x\/\\y in {-1\/1.5,-1.5\/1.3,-0.5\/1.7} \\draw[-,color=blue] (0,0) to (\\x,\\y);\n  \\foreach \\x\/\\y in {1\/1.5,1.5\/1.3, 1.8\/0.8} \\draw[-,color=blue] (0,0) to (\\x,\\y);\n  \\fill[black] (0,0) circle (2pt);\n  \\fill[black] (2,0) circle (2pt);\n    \\fill[black] (-2.2,1.1) circle (2pt);\n      \\fill[blue] (-1,1.5) circle (2pt);\n        \\fill[blue] (-1.5,1.3) circle (2pt);\n          \\fill[blue] (-0.5,1.7) circle (2pt);\n            \\fill[blue] (1,1.5) circle (2pt);\n              \\fill[blue] (1.5,1.3) circle (2pt);\n                \\fill[blue] (1.8,0.8) circle (2pt);\n                  \\fill[blue] (0,1.8) circle (2pt);\n\\node[above] at (-1,1.5) {};\n  \\node[above, red] at (-1.7,1.3) {$w$};\n   \\node[above] at (-1,1.5) {$w_2$};\n     \\node[above] at (1.7,1.3) {$w_i$};\n\\end{tikzpicture}\n\\end{center}\n\\end{minipage}\n\n\\noindent If $\\Gamma$ is not semiample, then\nany section of $R_w = H^0(C^{(2)},D)$\nis divisible by $f_1$, a contradiction.\nThus we showed that $\\Gamma$ is semiample\nand as a consequence of Lemma~\\ref{triv}\nthe curve $C$ has two $g_k^1$. We denote\nthe corresponding curves of $C^{(2)}$ by \n$\\Gamma$ and $\\Gamma'$.\nA multiple of $\\Gamma$\ndefines a morphism $f\\colon C^{(2)}\\to B$\nonto a smooth curve $B$ whose Stein factorization is \nthe following \n\\[\n \\xymatrix{\n C^{(2)}\\ar[r]^{\\varphi}\\ar[rd]_{f} & Y\\ar[d]\\\\\n & B.\n }\n\\]\nTwo fibers of $\\varphi$ are $n\\Gamma$ and $m\\Gamma'$\nfor some positive rational numbers $n,m$. Since\n$\\Gamma$ is numerically equivalent to $\\Gamma'$ \nby Lemma~\\ref{gamma}, then $n=m$. \nMoreover by Lemma~\\ref{fib} the curve $Y$\nis rational, so that $n\\Gamma$ is linearly equivalent\nto $n\\Gamma'$. This implies that the class\nof $\\Gamma-\\Gamma'$ is torsion non-trivial\nin $\\Pic^0(C^{(2)})$\nand thus the same holds for \n\\begin{align*}\n 2\\eta_C \n & =\n \\text{ramification of $g_k^1$ - ramification of ${g_k^1}'$}\\\\\n & =\n \\imath^*(\\Gamma-\\Gamma').\n\\end{align*}\nWe now show that $(ii)\\Rightarrow (i)$ holds.\nFirst of all observe that if $\\eta_C$ is torsion\nnon-trivial, then the same holds for\n$\\Gamma-\\Gamma'$ by the above equalities\nand the fact that $\\imath^*$ is an isomorphism.\nIn this case $n\\Gamma\\sim n\\Gamma'$ for\nsome positive integer $n$ and this implies\nthat $|n\\Gamma|$ is base point free, being\n$\\Gamma^2=0$. Thus $\\Gamma$ is semiample.\nHence, if $L\\subseteq{\\mathbb Z}^2$ is the submonoid\ngenerated by the integer points of the cone \n${\\rm cone}(2K+3\\Delta,2K+(5-2k)\\Delta)$,\nthe following subalgebra \n\\[\n  S := \\bigoplus_{(a,b)\\in L}H^0(C^{(2)},a\\Delta+bK)\n\\]\nof $R$ is finitely generated \nby~\\cite[Lemma 4.3.3.4]{ADHL}.\nThe homogeneous elements of $R$ \nwhich do not belong to $S$ are sections\nof Riemann-Roch spaces \n$H^0(C^{(2)},D)$ with $D\\cdot\\Delta < 0$.\nHence any such section is divisible by\nthe generator $f_\\Delta$ of $R$ which is a defining\nsection for $\\Delta$. Thus we conclude that \n$R$ is generated by $S$ and $f_\\Delta$\nand the statement follows.\n\\end{proof}\n\n\n \n\\begin{remark}\n\\label{ch}\nOver an algebraically closed field ${\\mathbb K}$ of positive \ncharacteristic the statement of Theorem~\\ref{teo-1}\nmust be modified as follows: the algebra\n$R(\\Delta,K)$ is finitely generated if and \nonly if $\\eta_C$ is torsion. Indeed if $\\eta_C$ \nis trivial, then ${\\mathcal O}_\\Gamma(\\Gamma)$ is trivial\nas well as shown in the proof of Lemma~\\ref{triv},\nthus $\\Gamma$ is semiample by~\\cite[Theorem 0.2]{Ke}.\nMoreover, if ${\\mathbb K}$ is the algebraic closure\nof a finite field, then $\\eta_C$ is always\ntorsion and thus $R(\\Delta,K)$ is always \nfinitely generated.\n\nThe conclusion of Lemma~\\ref{covering}\nis no longer true in positive characteristic. Indeed in\nthis case the algebraic fundamental group of the\naffine line ${\\mathbb K}$ is not trivial and thus there is no\ncontradiction. For example, in characteristic $p>0$,\nif $C$ is the curve of ${\\mathbb K}^2$ defined by the equation \n$x_1^p-x_1=x_2$, then the projection onto the second\nfactor defines a non-trivial \\'etale covering of ${\\mathbb K}$.\n\\end{remark}\n\n\\begin{remark}\nWe observe that the locus of smooth curves of genus\n$(k-1)^2 > 1$ which admit two $g_k^1$ has one component\nof maximal dimension which consists of curves of \ntype $(k,k)$ on a smooth quadric $Q$. \nIndeed by~\\cite{AC} the only\nother component which could have bigger dimension \nwould consist of curves $C$ admitting an involution. \nA parameter count shows that for our curves this \ncomponent has smaller dimension.\n\nMoreover any smooth curve $C$ of type\n$(k,k)$ on $Q$ admits exactly two $g_k^1$.\nIndeed let $S = \\{p_1,\\dots,p_k\\}$ be a set of\n$k$ distinct points with $p_1+\\dots+p_k\\in g_k^1$. \nBy the Riemann-Roch theorem $S$ is in Cayley-Bacharach \nconfiguration with respect to the curves of \ntype $(k-2,k-2)$. It follows that all the \npoints of $S$ are collinear. Indeed, let\n$\\ell$ be the line through the first two points\n$p_1,p_2$, let $H$ be a general hyperplane \nwhich contains $\\ell$ and let $q\\in S\\setminus\\{p_1,p_2\\}$.\nTake a union $\\Lambda$ of $k-3$\nhyperplanes through all the points of \n$S\\setminus\\{p_1,p_2,q\\}$ and such that\n$q\\notin\\Lambda$.\nThen $H\\cup \\Lambda$ cuts out on $Q$ \na curve of type\n$(k-2,k-2)$ which, by the Cayley-Bacharach\nconfiguration, must pass through $q$.\nHence $q\\in H$ and by the generality\nassumption on $H$ we deduce $q\\in\\ell$.\n\\end{remark}\n\n\n\n\\section{The grid family}\n\\label{grid-fam}\n\nIn this section we study families of curves of type $(k,k)$\non a smooth quadric $Q = {\\mathbb P}^1\\times{\\mathbb P}^1$\nsuch that the class of the difference of the two\n$g_k^1$ induced by the two rulings is \n$n$-torsion. We prove that $n\\geq k$\nand construct the family with $n=k$.\n\n\n\\begin{definition}\n\\label{grid}\nGiven two effective \ndivisors $L_1$ and $L_2$ of $Q$ of type\n$(k,0)$ and $(0,k)$ respectively, the {\\em grid \nlinear system} defined by $L_1$ and $L_2$ is \nthe linear system of curves of $Q$ of \nbi-degree $(k,k)$ which pass through the \ncomplete intersection $L_1\\cap L_2$.\nThe {\\em grid family} \n\\[\n \\mathcal G_k\n \\subseteq\\mathcal F_k\n\\]\nis the family \nof all curves of type $(k,k)$ which belong to some \ngrid linear system.\n\\end{definition}\nObserve that if $C$ is a smooth curve in $\\mathcal G_k$\nand $\\eta_C\\in \\Pic^0(C)$ is the class of the difference of the \ntwo $g_k^1$ cut out by the two rulings then $\\eta_C^{\\otimes k}$\nis trivial. This justifies the inclusion\n$\\mathcal G_k\\subseteq\\mathcal F_k^{\\rm tor}$.\nAny curve $C$ in $\\mathcal G_k$\nadmits an equation of the form\n\\begin{equation}\n\\label{equ-fam}\n f_1(x_0,x_1)g_2(y_0,y_1)+g_1(x_0,x_1)f_2(y_0,y_1)\n =\n 0,\n\\end{equation}\nwhere $f_1, f_2, g_1$ and $g_2$ are homogeneous\npolynomials of degree $k$. Indeed it is enough to\nprove the claim for a curve $C$ in a grid family\nwhere both $L_1$ and $L_2$ consist of\ndistinct lines and then conclude by specialization \nthat the same holds for any\ncurve $C$ of $\\mathcal G_k$.\nLet $h=0$ be an equation for $C$ and\nlet $f_i = 0$ be an equation for $L_i$, for $i=1,2$.\nBy the equality\n\\[\n V(h,f_1) = V(f_1,f_2),\n\\]\nthe fact that the two ideals $(h,f_1)$ and $(f_1,f_2)$ are both\nradical and saturated with respect to the\nirrelevant ideal $(x_0,x_1)\\cap (y_0,y_1)$ of $Q$,\nwe deduce that the equality $(h,f_1) = (f_1,f_2)$\nholds. The claim follows since $h$ has bi-degree $(k,k)$.\nObserve that the general element of $\\mathcal G_k$ is\nirreducible and smooth.\n\n\\begin{proposition}\nThe {\\em grid family} $\\mathcal G_k$\nhas dimension $4k-1$.\n\\end{proposition}\n\\begin{proof}\n\nThe projectivization of the set of homogeneous polynomials of\nbidegree $(k,k)$ of type $f(x_0,x_1)g(y_0,y_1)$ can be identified \nwith the image $\\mathcal S$ of the Segre embedding of \n${\\mathbb P}^k\\times{\\mathbb P}^k\\to{\\mathbb P}^N$,\nwhere $N={k^2+2k}$.\nThus, a curve in $\\mathcal G_k$ can be identified \nwith a point of the $1$-secant variety ${\\rm Sec}(\\mathcal S)$\nof $\\mathcal S$.\nBy~\\cite[Theorem 1.4]{CS} the dimension of\n${\\rm Sec}(\\mathcal S)$ is $4k-1$ and the \nstatement follows.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{composed}\nLet $C\\in\\mathcal F_k$ and let $D$ be a divisor\nof $C$ cut out by one of the two rulings. Then,\nfor any $1\\leq n\\leq k-1$, the linear system $|nD|$ \nis composed with the pencil $|D|$.\n\\end{lemma}\n\\begin{proof}\nTo prove the statement it is enough to show that\n$h^0(C,nD) = n+1$ for $1\\leq n\\leq k-1$. \nLet $Q = {\\mathbb P}^1\\times{\\mathbb P}^1$. Without loss of generality\nwe can assume that $D$ is cut out by the first ruling\nof $Q$, so that we have the following exact sequence \nof sheaves\n\\[\n \\xymatrix@1{\n  0\\ar[r] \n  &\n  {\\mathcal O}_Q(-k+n,-k)\\ar[r]\n  &\n  {\\mathcal O}_Q(n,0)\\ar[r]\n  &\n  {\\mathcal O}_C(nD)\\ar[r]\n  &\n  0.\n }\n\\]\nTaking cohomology, using the vanishing of the higher\ncohomology groups of the middle sheaf, the Serre's \nduality theorem and the hypothesis on $n$ we deduce \nthe following equalities: \n\\[\nh^1(C,nD) = h^0(Q,{\\mathcal O}_Q(k-n-2,k-2))= (k-n-1)(k-1).\n\\]\nBy the adjunction formula $C$ has genus $(k-1)^2$.\nThus by the above and the Riemann-Roch formula \nwe conclude\n\\[\n h^0(C,nD)\n =\n nk+1-(k-1)^2+(k-n-1)(k-1)\n =\n n+1\n\\]\nand the statement follows.\n\\end{proof}\n\n\\begin{proposition}\n\\label{torsion}\nLet $C$ be a smooth curve of  type\n$(k,k)$ on $Q$ and let $\\eta_C$\nbe the class of the difference of the two $g_k^1$. \nThen $\\eta_C$ has order $\\geq k$ and the \nfollowing are equivalent:\n\\begin{enumerate}\n\\item\n$\\eta_C$ has order $k$;\n\\item\nthe curve $C$ belongs to the grid family $\\mathcal G_k$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\nWe first show that $\\eta_C$ has order $\\geq k$.\nLet $D_1$ be a divisor in the first $g_k^1$ and let  \n$D_2$ be a divisor in the second $g_k^1$. \nLet $n<k$ be a positive integer.\nBy Lemma~\\ref{composed} the linear system\n$|nD_i|$ is composed with $|D_i|$ for any $i$. \nThus the linear equivalence $nD_1\\sim nD_2$ would \nimply that for any set of $n$ lines of the first ruling \nthere are $n$ lines of the second ruling which cut \nout the same set of points on $C$.  This cannot be\nsince such set of $nk$ points would lie on a grid \nwith $n^2$ points.\nThis proves the claim.\n\nWe now show that $(i)\\Rightarrow (ii)$ holds.\nBy the exact sequence of sheaves\n\\[\n \\xymatrix@1{\n  0\\ar[r] \n  &\n  {\\mathcal O}_Q(0,-k)\\ar[r]\n  &\n  {\\mathcal O}_Q(k,0)\\ar[r]\n  &\n  {\\mathcal O}_C(kD_1)\\ar[r]\n  &\n  0\n }\n\\]\nthe fact that $h^0 = 0$, $h^1=k-1$ for the first sheaf\nand $h^0 = k+1$, $h^1 = 0$ for the second sheaf  we\nget the equality $h^0(C,{\\mathcal O}_C(kD_1)) = 2k$. \nBy hypothesis $|kD_1| = |kD_2|$ holds and by the above \ncalculation the linear system is a projective space of dimension\n$2k-1$. Let $\\mathcal H_i\\subseteq |kD_1|\\cong{\\mathbb P}^{2k-1}$ \nbe the projectivization of the $k$-th symmetric\npower of the vector space $H^0(C,D_i)$.\nSince both $\\mathcal H_1$ and $\\mathcal H_2$\nhave dimension $k$, their intersection \n$\\mathcal H_1\\cap\\mathcal H_2$ is at least\none-dimensional. A point of this intersection \ncorresponds to a divisor $D$ which is cut out \non $C$ by $k$ lines of the first ruling\nand by $k$ lines of the second ruling. \nThis proves the claim.\n\nThe implication $(ii)\\Rightarrow (i)$ is obvious.\n\\end{proof}\n\n\n\n\\section{Density of the torsion locus}\n\\label{density}\n\nWe define $\\mathcal F_k$ \nto be the open subset of the Hilbert scheme of the smooth\ncurves of type $(k,k)$ of ${\\mathbb P}^1\\times{\\mathbb P}^1$.\nLet $\\pi\\colon \\mathcal C\\to\\mathcal F_k$ be the universal\nfamily, let $\\mathcal H = R^1\\pi_*{\\mathbb C}$ be the associated\nHodge bundle whose fiber\nover a point $C\\in\\mathcal F_k$ is the\ncohomology group $H^1(C,{\\mathbb C})$ and\nlet $\\mathcal H^{1,0}$ be the subbundle \nof $\\mathcal H$ whose fiber over\n$C$ is $H^{1,0}(C)$. We recall that\nthe jacobian family $\\mathcal J\\to\\mathcal F_k$\ncan be defined as \n\\begin{equation}\n\\label{jac}\n \\mathcal J\n =\n \\frac{\\mathcal H}{\\mathcal H^{1,0}+R^1\\pi_*{\\mathbb Z}}.\n\\end{equation}\nGiven a point $t\\in\\mathcal F_k$ we denote by\n$C_t$ the fiber of $\\pi$ over $t$ and \nby $\\nu_t = \\mathcal L_1\\otimes\\mathcal L_2^{-1}\n\\in J(C_t)$ the class of the difference \nof the two $g_k^1$ cut out by the two rulings.\nThis gives a normal function $\\nu\\colon\\mathcal F_k\n\\to \\mathcal J$ defined by $t\\mapsto\\nu_t$.\nWe now consider a $\\mathcal C^\\infty$ trivialization \n$\\varphi$\nof the jacobian family over an open, simply connected\nsubset $\\mathcal U$ of $\\mathcal F_k$:\n\\[\n \\xymatrix{\n \\mathcal J\n  \\ar[r]^-\\varphi & \n  \\mathcal U\\times\\dfrac {R^1\\pi_\\ast\\mathbb R} {R^1\\pi_\\ast \\mathbb Z}\n  = \n  \\mathcal U\\times\\mathbb T,\n }\n\\]\nwhere $\\mathbb T$ is the real torus \n${\\mathbb R}^{2g}\/{\\mathbb Z}^{2g}$.\nWe let ${\\rm pr}_2\\colon\\mathcal U\\times\\mathbb T\n\\to\\mathbb T$ be the projection on the second factor\nand let \n\\[\n \\eta\\colon{\\mathcal  U}\\to\\mathbb T\n \\qquad\n t\\mapsto {\\rm pr}_2(\\varphi(\\nu(t))).\n\\]\nWe will show that, for any $k\\geq 3$, the image \nof $\\eta$ contains a non-empty open subset.\nOur strategy will be to prove that \nthe fibers of $\\eta$ are complex subvarieties \nof $\\mathcal U$, that $\\eta$ has a fiber \nof the expected real dimension \n$8k-2 = \\dim \\mathcal F_k - \\dim\\mathbb T$\nand we conclude by Proposition~\\ref{fiber}\n(see also the argument in the proof of~\\cite[Proposition 3.4]{CPP}).\n\n\\begin{proposition}\n\\label{fiber}\nLet $V\\subseteq X$ be an inclusion of complex manifolds, \n$Y$ be a real manifold and \n$\\eta\\colon X\\to Y$ be a $\\mathcal C^\\infty$\nmap whose fibers are complex subvarieties\nof $X$. Assume that there is a point $p\\in V$\nsuch that the following hold:\n\\begin{enumerate}\n\\item\nthe differential $d\\eta_p$ is surjective,\n\\item\nthe fiber of the restriction $\\eta|_V$ over $\\eta(p)$\nhas dimension $\\dim V - \\dim Y\\geq 0$.\n\\end{enumerate}\nThen $\\eta(V)$ contains\na non-empty open subset of $Y$.\n\\end{proposition}\n\\begin{proof}\nLet $F$ be the fiber of $\\eta$ over $\\eta(p)$ \nand let $U$ be a coordinate neighbourhood of $p$\nsuch that the restriction $\\eta|_U$ is a submersion.\nLet $d = \\dim V - \\dim Y$. After possibly \nintersecting $V$ with $d$ general smooth \ncomplex hypersurfaces passing through $p$\nwe can reduce to the case $d = 0$.\nThus, after possibly shrinking $U$ \nwe can assume $F\\cap V\\cap U = \\{p\\}$.\nFor $q$ in a sufficiently small neighbourhood\n$W$ of $\\eta(p)$ the intersection number \n$\\eta^{-1}(q)\\cap V$ does not change\nby~\\cite[Pag.\\,664]{GH}, so that the intersection is\nnon-empty. Then $\\eta(V)$ contains $W$\nand the statement follows.\n\\end{proof}\n\n\\begin{proposition}\nThe fibers of the map $\\eta$ are complex varieties.\n\\end{proposition}\n\\begin{proof}\nWe restrict the Hodge and the jacobian bundle \nto the simply connected open subset $\\mathcal U$\nof $\\mathcal F_k$,\nwhere the family is topologically trivial.\nLet $t_0\\in \\mathcal U$ be a distinguished  point\nand let $C = C_{t_0}$ be the corresponding\ncurve. We construct the following\ncommutative diagram\n\\[\n \\xymatrix{\n  \\mathcal U\\ar@{=}[d]\\ar[r]^-{\\tilde\\nu}\n  & \\mathcal H_{\\mathcal U}\\ar[r]^-\\cong\\ar[d]\n  & \\mathcal U\\times H^1(C,{\\mathbb C})\\ar[d]\\ar[r]\n  & H^1(C,{\\mathbb C})\\ar[d]\\\\\n  \\mathcal U\\ar[r]^-{\\nu|_\\mathcal U}\n  & \\mathcal J_{\\mathcal U}\\ar[r]^-\\varphi\n  & \\mathcal U\\times\\mathbb T\\ar[r]\n  & \\mathbb T\n }\n\\]\nwhere $\\tilde\\nu$ is a lifting of the restriction\n$\\nu|_{\\mathcal U}$ and the maps in the top row\nare complex analytic morphisms.\nObserve that, given $x\\in \\mathbb T$, we have \n\\[\n\\eta^{-1}(x)=\\{t\\in\\mathcal U: \\tilde \\nu(t)-\\tilde x+c\\in\\mathcal H^{1,0}\\},\n\\]\nwhere the projection of $\\tilde x$ to $\\mathcal J$ is $x$ \nand $c\\in H^1(C,{\\mathbb Z})$.\nThis implies that $\\eta^{-1}(x)$ \nis a complex subvariety of $\\mathcal U$ \nsince $\\mathcal H^{1,0}$ is a \ncomplex subbundle of $\\mathcal H$.\n\\end{proof}\n\n \n \n\\begin{proof}[Proof of Theorem~\\ref{teo-2}]\nWe will start proving that the image of the map $\\eta$ \ncontains a non-empty open subset of $\\mathbb T$.\nWe consider the map\n\\[\n \\phi\\colon\\mathcal J_{\\mathcal U}\\to \\mathbb T\n\\]\nwhich is composition of the trivialization $\\varphi$\nwith the projection onto the second factor.\nObserve that the differential of $\\phi$ has\nmaximal rank at any point. By applying\nProposition~\\ref{fiber} to the map\n$\\phi$, the subvariety \n$V = \\nu|_{\\mathcal U}(\\mathcal U)$ of $\\mathcal J$\nand the fiber $\\mathcal G_k$ of the restriction\n$\\phi|_V$ we deduce that $\\eta(V)$ contains\na non-empty open subset.\nObserve that the trivialization $\\varphi$\ncan be taken real analytic, since it is\nobtained by taking the real part of the \ncomplex analytic isomorphism coming\nfrom the trivialization of the Hodge bundle.\nBy Sard theorem the set of non-critical values\nof $\\phi$ is an open subset in the analytic \ntopology. Since $\\phi$ admits an analytic continuation\non any simple arc in $\\mathcal J$, its set\nof non-critical points is dense in the analytic\ntopology.\nWe conclude by observing that the set\nof torsion points of $\\mathbb T$ is dense\nand $\\mathcal F_k^{\\rm tor}$ is the preimage\nof this set via $\\eta = \\phi\\circ\\nu$.\n\\end{proof}\n\n\n\\section{Hyperelliptic curves}\n\\label{hyp}\n\nIn this section we prove a density\ntheorem for hyperelliptic curves. This result \nhas an independent interest and is proved in \nthe spirit of Griffiths computations of the \ninfinitesimal invariant~\\cite{Gr}.\nAs an application to our problem, this result\nprovides an alternative proof for Theorem~\\ref{teo-2}\nin case $g=4$.\n\nLet $\\pi\\colon\\mathcal C\\to\\mathcal U$ be a versal family \nof hyperelliptic curves of genus $g > 1$, where $\\mathcal U$\nis simply connected of dimension $2g-1$.\nWe denote by $j\\in\\Aut(\\mathcal C)$ the hyperelliptic\ninvolution and by\n$\\mathcal J\\to\\mathcal U$ the jacobian\nfamily. We consider the Abel-Jacobi map \n\\[\n \\nu\\colon \\mathcal C \\to \\mathcal J,\\quad  \n x\\mapsto\\int_{j(x)}^x.\n\\]\nIf we take $\\pi^\\ast \\mathcal J\\to \\mathcal C$ the pull-back \nof the Jacobian family on $\\mathcal C,$ \nwe may consider $\\nu$ as a normal function.\nAs in Section~\\ref{density} we consider a $\\mathcal C^\\infty$\ntrivialization of the jacobian family\n$\\mathcal J\\cong \\mathcal U\\times \\mathbb T$,\nwhere $\\mathbb T\\cong J(C)$, to construct a map\n\\[\n \\gamma\\colon\\mathcal C\\to\\mathbb T.\n\\]\n\\begin{theorem}\n\\label{diff}\nIf $p\\in C$ is not a Weierstrass point, then the\ndifferential of $\\gamma$ at $p$ is surjective.\n\\end{theorem}\n\nOur strategy is as follows. For any holomorphic\nform $\\omega\\in H^0(C,\\Omega_C)$ we show that\nthere is a curve $r(t)$ in $\\mathcal C$ such that\n$r(0) = p$ and $d\\gamma_p(r'(0))\\cdot\\omega$\nis non-zero. To this aim we produce $r(t)$\naccordingly to the order $n$ of vanishing of $w$ \nat $p$. Since the divisor ${\\rm div}(w)$ is $j$-invariant, \nthen it is natural to consider $D = p+j(p)$.\nWe thus have a filtration \n\\begin{equation}\n\\label{filt}\n H^0(C,\\Omega_C)\n =\n L^0\\supseteq L^1\\supseteq...\\supseteq L^{g-1}\\supseteq L^g=0,\n\\end{equation}\nwhere $L^k$ is the Riemann-Roch space \n$H^0(C,\\Omega_C(-kD))$. Given $\\omega\\in L^k\n\\setminus L^{k+1}$, with $k>0$, we construct $\\zeta = \\partial(f)\n\\in H^1(C,T_C)^j$ as in Subsection~\\ref{def}, where $f\\in\nH^0(Z,{\\mathcal O}_Z)$. The one-dimensional family\n\\[\n \\mathcal C_\\zeta\\to\\Delta\n\\]\ndefined by $\\zeta$, plus a choice of a smooth\nsection through the point $p$ in the family, defines a curve \n$r(t)$ in $\\mathcal C$. We show that the family\nis equipped with a $\\mathcal C^\\infty$\n $1$-form $\\Theta$ such that the restriction \nof the $(1,0)$-part $\\Theta^{1,0}$\nto the central fiber $C$\nadmits a local expansion at $p$ of the\nform $w + \\tilde f(z)dt + o(t)$, where $z$ is a coordinate\nin $C$, $\\tilde f|_Z=f$ and $t\\in\\Delta$. We finally prove \n\\[\n d\\gamma_p(r'(0))\\cdot\\omega\n =\n \\lim_{t\\to 0}\\frac{1}{t}\\left(\\int_{\\Gamma_t}\\Theta_t-\\int_{\\Gamma_0}\\omega\\right)\n =\n 2f(p)\\neq 0,\n\\]\nwhere $\\Gamma_t$ is a path between $r(t)$ \nand $j(r(t))$. When $k=0$ we choose $r(t)$\nto be a path within the central fiber $C$,\nwe write $\\omega$ locally as $h(z)dz$\nand prove the equality\n\\[\n d\\gamma_p(r'(0))\\cdot\\omega\n =\n \\lim_{t\\to 0}\\frac{1}{t}\\int_{p}^{r(t)}\\omega\n =\n h(0)\\neq 0.\n\\]\n\n\n\\subsection{Deformation of curves}\n\\label{def}\nWe recall first a result on deformation and on \nextension of line bundles \nwhich will be applied to hyperelliptic curves (see also~\\cite{CP} and~\\cite{R}).\nLet $C$ be a smooth curve of genus $g>1$ and \nlet $T_C$ and $\\Omega_C$ be respectively the holomorphic \ntangent bundle and the canonical line bundle of $C$. \nFix a non trivial $\\omega\\in H^0(C,\\Omega_C)$ and \nlet $Z$ be the canonical divisor associated to $\\omega.$\nThe form $\\omega$ defines the following exact sequence\n\\[\n\\xymatrix@1{\n0\\ar[r] & T_C\\ar[r]^{\\omega}& {\\mathcal O}_C\\ar[r]& {\\mathcal O}_Z\\ar[r]& 0.\n}\n\\]\nPassing to the long exact sequence in cohomology \nwe obtain\n\\[\n\\xymatrix@C=20pt{\n0\\ar[r]& {\\mathbb C}\\cong H^0(C,{\\mathcal O}_C)\\ar[r]& \nH^0(Z,{\\mathcal O}_Z)\\ar[r]^\\partial& H^1(C,T_C)\n\\ar[r]^{\\omega}& H^1(C,{\\mathcal O}_C).\n}\n\\]\nGiven an element $f\\in H^0(Z,{\\mathcal O}_Z)$\nits image $\\zeta=\\partial(f)$ defines an\nextension of ${\\mathcal O}_C$ by $T_C$ via the\nisomorphism $H^1(C,T_C)\\cong \n{\\rm Ext}^1({\\mathcal O}_C,T_C)$\n\\[\n\\xymatrix@1{\n0\\ar[r] & T_C \\ar[r] & E_{\\zeta}\\ar[r] & {\\mathcal O}_C\\ar[r] & 0.\n}\n\\]\nTaking tensor product with $\\Omega_C$ \nand recalling that $T_C$ is dual with $\\Omega_C$\nwe get the following exact sequence\n\\begin{equation} \\label{zeta}\n\\xymatrix@1{\n0\\ar[r] &{\\mathcal O}_C\\ar[r] & E_{\\zeta}\\otimes \\Omega_C\\ar[r] &\n \\Omega_C\\ar[r]& 0,\n }\n \\end{equation}\nwhich passing to the long exact sequence in cohomology\ngives the following sequence \nwhose coboundary  \nis the cup product with $\\zeta$:\n\\[\n \\xymatrix@C=20pt{\n0\\ar[r] & {\\mathbb C}\\cong H^0(C,{\\mathcal O}_C)\\ar[r]&\n H^0(C,E_\\zeta\\otimes \\Omega_C)\\ar[r]^-k & \n H^0(C,\\Omega_C)\\ar[r]^-{\\zeta}\\ar[r]&\n  H^1(C,{\\mathcal O}_C).\n  }\n\\]\nSince $\\zeta\\in\\ker(\\omega)$, or equivalently\nthe cup product $\\zeta\\cdot \\omega$ vanishes,\nthere exists an element $\\Omega\\in \nH^0(C,E_\\zeta\\otimes \\Omega_C)$ such that \n$k(\\Omega)=\\omega$.\nNow we consider the commutative diagram\n\n\\[\n \\xymatrix@1{\n  & 0\\ar[d] & 0\\ar[d] & H^0(C,{\\mathcal O}_C)\\ar[d]^-\\zeta\\\\\n  0\\ar[r]\\ar[d]\n  & H^0(C,{\\mathcal O}_C)\\ar[r]\\ar[d] \n  & H^0(Z,{\\mathcal O}_Z)\\ar[d]^-\\rho\\ar[r]^-\\partial \n  & H^1(C,T_C)\\ar[d] \\\\\n  0\\ar[r]\\ar[d]\n  & H^0(C,E_\\zeta\\otimes \\Omega_C)\\ar[r]^-r\\ar[d]^-k\n  & H^0(Z,E_\\zeta\\otimes \\Omega_C|_Z)\\ar[d]\\ar[r]\n  & H^1(C,E_\\zeta)\\ar[d]\\\\\n  H^0(C,{\\mathcal O}_C)\\ar[r]^-\\omega\n  & H^0(C,\\Omega_C)\\ar[r]\n  & H^0(Z,\\Omega_C|_Z)\\ar[r]\n  & H^1(C,{\\mathcal O}_C)\n }\n\\]\n \\vspace{0.2cm}\n \nThe restriction of the lifting $\\Omega$ to $Z$ \ngives an element $r(\\Omega)\\in H^0(Z,E_\\zeta\\otimes \\Omega_C|_Z)$ \nthat by construction is in the image of the map\n$\\rho$, that is $\\Omega=\\rho(g)$ \nfor some $g\\in H^0(Z,{\\mathcal O}_Z)$.\nA diagram chase proves indeed that\n$\\partial (g)=\\zeta$ holds.\nSince $\\ker(\\partial)$ is isomorphic to ${\\mathbb C}$,\nwe conclude that $f$ equals $g$ up to a constant.\nThis means that we can realize the function $f$ \nby a unique suitable lifting $\\Omega$ of $\\omega.$\nWe collect the discussion in the following lemma.\n\n\\begin{proposition}\\label{coord}\nLet $\\omega$ be a non-zero element in  \n$H^0(C,\\Omega_C)$ and let $Z$ be the \ndivisor of $\\omega$. \nThen for any $f\\in H^0(Z,{\\mathcal O}_Z)$ \nthere is a unique \n$\\Omega\\in H^0(C,E_{\\zeta}\\otimes \\Omega_C)$ \nsuch that $r(\\Omega)=\\rho (f)$. \n\\end{proposition}\n\nWhen we interpret $H^1(C,T_C)$ \nas the space of first order deformations of $C$,\nso that $\\zeta$ corresponds to a family\n\\[\n\\mathcal C_\\zeta\\to {\\rm Spec}\\, {\\mathbb C}[\\varepsilon],\n\\]\nthe isomorphism $H^1(C,T_C)\\to\n{\\rm Ext}^1({\\mathcal O}_C,\\Omega_C)$\ngives the identifications \n$E_{\\zeta}\\cong T_{\\mathcal C}|_{C}$ \nand $E_{\\zeta}\\otimes \\Omega_C\\cong\\Omega_{\\mathcal C}|_{C}.$ \nTherefore the sequence~\\eqref{zeta}\nis the cotangent sequence of the first order deformation.\nIn coordinates we may write \n\\begin{equation}\n\\label{Omega}\n \\omega= h(z)dz,\\qquad \\Omega=h(z)dz+\\tilde f(z)dt,\n\\end{equation}\nwhere $dt$ is the global section \nof the cotangent $\\Omega_{\\mathcal C}|_{C}$ \nand $f=\\tilde f|_Z$ is a section of $H^0(Z,{\\mathcal O}_Z)$ \nsuch that $r(\\Omega)=\\rho (f)$.\n\n\n\\subsection{The normal function}\nIn this subsection we will specialize the previous\nconstruction to hyperelliptic families.\nFirst of all, given a hyperelliptic curve $C$ of genus $g>1$,\nwe consider the $j$-invariant subspace\n\\[\n H^1(C,T_C)^{j}\n \\subseteq\n H^1(C,T_C),\n\\]\nwhich corresponds to the directions that are preserved\nby the hyperelliptic involution $j$.\nSince $j$ acts on $H^0(Z,{\\mathcal O}_Z)$\nas $f\\mapsto f\\circ j$\nand it acts as $-1$ on $H^0(C,\\Omega_C)$, \nthen $\\partial(f) = \\zeta$ is $j$-invariant if and only if\n$f\\circ j$ equals $-f$ up to a constant.\n\nLet $p$ be a non-Weierstrass point of $C$ and\n$\\omega\\in H^0(C,\\Omega_C)$\nbe a holomorphic form which vanishes \nwith order $k>0$ at $p$,\nthat is $\\omega\\in L^k\\setminus L^{k+1}$.\nWe now take $f_{\\omega}\\in H^0(Z,{\\mathcal O}_Z)$, where $Z \n= k(p+j(p)) + Z' = {\\rm div}(\\omega)$,\nsuch that\n\\[\n f_{\\omega}(p)=1,\\quad f_{\\omega}(j(p))=-1,\\quad f_{\\omega}(p')=0 \\text{ for } p'\\in Z'.\n\\]\nGiven $\\zeta_{\\omega}=\\partial(f_{\\omega})$,\nby the previous remark\nwe have $j(\\zeta_{\\omega})=\\zeta_{\\omega}.$ \nThus there exists  a smooth family of hyperelliptic curves\n\\begin{equation}\n \\pi\\colon \\mathcal C_\\omega\\to \\Delta \\label{curvamod},\n\\end{equation} \n where $\\Delta\\subset \\mathcal U$ is a disk, such that \n$\\pi^{-1}(0)=C$ and such that \nthe Kodaira-Spencer \nclass of the family is $\\zeta_{\\omega}$. \nLet $\\Omega_{\\omega}$ be the section of \n$E_{\\zeta_{\\omega}}\\otimes \\Omega_C$ \nassociated to $f_{\\omega}$ as in Lemma ~\\ref{coord}.\n By means of a trivialization of \n the family we can construct a closed \n differential $1$-form $\\Theta$\n on $\\mathcal C_{\\omega}$ which is invariant \n with respect to the involution $j$ and\n such that the restriction of the $(1,0)$-part\n $\\Theta^{(1,0)}$ to the central fiber equals \n $\\Omega_{\\omega}$. \nIn local coordinates we can write  \n  \\[\n  \\Theta(z,t)^{(1,0)}=\\Omega_{\\omega}+ o(t)=\\omega+f_{\\omega}(z)dt+o(t).\n  \\]\n\nWe also assume to have a holomorphic section $r$ \nof $\\pi_{\\omega}$ such that $r(0)=p$ and define $r'= j(r).$\nMoreover, we fix a differentiable map\n\\[\n\\Gamma(t,s):\\Delta\\times [0,1]\\to \\mathcal C_{\\omega}\n\\] \nsuch that $\\Gamma(t,s)\\in \\pi^{-1}(t)$,\n $\\Gamma(t,0)=r'(t)$ and  $\\Gamma(t,1)=r(t).$ \n Observe that $\\Gamma$ is a family of sections \n connecting $r'$ and $r$.\n We define the function\n\\[\n g\\colon\\Delta\\to {\\mathbb C},\\quad \n t\\mapsto \n \\int_{\\Gamma_t} \\Theta_{t}.\n\\]\n Following Griffiths~\\cite[(6.6)]{Gr} and using the fact that \n the Gauss-Manin connection vanishes on $\\Theta$,\n we have that the derivative of $g$ at $0$ \n equals\n$\nd\\gamma_p(r'(0))\\cdot \\omega.\n$\n \n\n\\begin{proof}[Proof of Theorem~\\ref{diff}]\nWith the previous notation, \ngiven any $\\omega\\in H^0(C,\\Omega_C)$ \nvanishing of order $k>0$ at $p$, \nwe consider the family $\\pi_{\\omega}$, \nwith its sections $r=r_{\\omega}$ and $r'=j(r)$, \nand $\\Theta$ the corresponding differential form on \n$\\mathcal C_{\\omega}$.\nBy the previous remark we have that \n\\[\nd\\gamma_p(r'(0))\\cdot \\omega=g'(0).\n\\]\nWe now compute the latter term:\n\\[\ng(t)-g(0)=\\int_{\\Gamma_t} \\Theta_{t}-\\int_{\\Gamma_0} \n\\Theta_{0}=\\int_{\\Gamma_t} \\Theta_{t}-\\int_{\\Gamma_0} \\omega.\n\\]\nWe call $r_t$ and $r'_t$ \nthe arcs $r([0,t])$ and $r'([0,t])$ respectively.\nSince $\\Theta$ is closed, then $\\Gamma^\\ast(\\Theta)$ is\nexact and we have \n$0= \\int_{r'_t}\\Theta_t+\\int_{\\Gamma_t}\\Theta_t - \\int_{r_t}\\Theta_t-\\int_{\\Gamma_0}\\Theta_t$, \nhence\n\\[\ng(t)-g(0)=  \\int_{r_t}\\Theta_t- \\int_{r'_t}\\Theta_t=2\\int_{r_t}\\Theta_t,\n\\]  \nwhere the last equality is due to the fact \nthat $j^\\ast(\\Theta)=-\\Theta.$\nFinally, since $\\Theta = \\omega+f_{\\omega}(z)dt + o(t)$,\nby the fundamental theorem of calculus \nwe get\n\\[\n \\lim_{t\\to 0}\\frac{1}{t}\\int_{r_t}\\Theta_t\n =\n \\lim_{t\\to 0}\\frac{1}{t}\\int_{r_t}\\Theta_t^{(1,0)}\n =\n f_{\\omega}(p)\\not=0.\n\\]\nThus $g'(0)\\not=0$.\nIf $k=0$, that is $\\omega$ does not vanish \nat $p$, we will choose a loop $r(t)$ in $C$ with\n$r(0)=p$ \nand we will compute the derivative \nof the Abel-Jacobi map ${\\rm AJ}$ on $C$. \nFirst note that if we take a \nWeierstrass point $q$ of $C$ \nwe have \n\\[\n{\\rm AJ}(r(t)-j(r(t)))=2{\\rm AJ}(r(t)-q).\n\\]\nTake a coordinate\n$z$ centered at $p$  \nsuch that $\\omega(z)=h(z)dz$ with $h(0)\\neq 0.$ \nFix a loop $r(t)$ such that $z(r(t))=t$, then\n\\[\n\\lim_{t\\to0}\\frac{1}{t} \\int_{q}^{r(t)}\\omega= \n\\lim_{t\\to 0}\\frac{1}{t}\\int_0^th(z)dz=h(0)\n\\] \nand we complete our result.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor}\nThe locus of curves $C$ in $\\mathcal M_4$ \nsuch that $\\eta_C$ is a non-trivial torsion point is a \ncountable union of subvarieties \nof complex dimension $\\geq 5$ \nand the set of subvarieties of\ndimension $5$ is dense in $\\mathcal M_4$\nin the analytic topology.\n\\end{corollary}\n\n\\begin{proof}\nLet $\\mathcal U$ be an open subset of $\\mathcal M_4$\nwhich intersects the hyperelliptic locus and let $\\tilde{\\mathcal U}$ \nbe the moduli space of pairs $(C,g_3^1)$,\nwhere $[C]\\in \\mathcal U$. \nLet $\\imath\\colon\\mathcal H\\to\\tilde{\\mathcal U}$ be the \nsubvariety containing pairs where $C$ is hyperelliptic. \nGiven a universal family \n$\\pi: \\mathcal C\\to  \\tilde{\\mathcal U}$,\nwe construct the following commutative diagram\n \\[\n \\xymatrix{\n  \\imath^*\\mathcal C\\ar[rrr]^-{(C,p)\\mapsto [j(p)-p]}\\ar[d]\n  &&& \\mathcal J\\ar@{=}[d]\\ar[rd]\\\\\n  \\mathcal C\\ar[d]^-\\pi\\ar[rrr]^-{(C,g_3^1)\\mapsto \\eta_C} \n  &&& \\mathcal J\\ar[r]^-{\\varphi} \n  &  \\tilde{\\mathcal U}\\times\n      \\mathbb T \\ar[d]^-{{\\rm pr}_2}\\\\\n \\tilde{\\mathcal U}\n  \\ar[rrrr]^-{[C]\\mapsto{\\rm pr}_2(\\varphi(\\eta_C))}\n  &&&& \\mathbb T\n }\n\\]\nwhere $\\eta_C={g'}_3^1-g_3^1=K_C-2g_3^1$ and $\\varphi$ is a $\\mathcal C^\\infty$-trivialization\n(defined after possibly shrinking $\\mathcal U$).\nThe commutativity of the top square comes from\nthe fact that on a hyperelliptic curve $C$ we have\n$g_3^1 = p + g_2^1$ and $K_C - 2g_3^1 = j(p)-p$,\nwhere $j\\in{\\rm Aut}(C)$ is the hyperelliptic involution.\nBy Theorem~\\ref{diff} the differential \nof  the map $\\gamma\\colon\\imath^*\\mathcal C\\to \\mathbb T$ \nobtained composing the maps in the diagram \nis surjective at any point $p$ which is not Weierstrass. \nThis implies that the map $\\eta: \\tilde{\\mathcal U}\\to \\mathbb T$ \nis locally a submersion at any point corresponding to a hyperelliptic \ncurve, in particular its image contains an open subset of $\\mathbb T$.\nWe thus conclude as in the last part of the \nproof of Theorem~\\ref{teo-2} given in section~\\ref{density}.\n\\end{proof}\n\n\n\\section{Examples}\n\\label{exa}\n\nIn this section we will provide further examples \nof curves having two $g_k^1$'s whose \ndifference is a torsion element in the Jacobian.\nIn particular we will show how to use automorphism\ngroups to construct new examples (see\nExample~\\ref{exa:new}).\n\n\n\\begin{proposition}\\label{auto}\nLet $C$ be a curve in $\\mathcal F_k$.\nIf $G$ is an automorphism group  of $C$\nof order $n$ which preserves each \n$g_k^1$ of $C$ and such that $C\/G$ \nhas genus zero, then the order of \n$\\eta_C$ divides $n$.\n\\end{proposition}\n\\begin{proof}\nLet $\\pi:C\\to C\/G\\cong {\\mathbb P}^1$ be \nthe quotient morphism, let \n$D=p_1+p_2+\\dots+p_k$ be an \nelement of the first $g_k^1$\nand let $q_i=\\pi(p_i)$. Then\nthe following linear equivalences\nhold\n\\[\n nD\n \\sim\n \\sum_{\\sigma\\in G} \\sigma^*(D)\n =\n \\pi^*(q_1)+\\pi^*(q_2)+\\dots+\\pi^*(q_k)\n \\sim\n kF, \n\\]\nwhere $F$ is a fiber of $\\pi$ and the first equivalence \nis due to the fact that $G$ preserves the \nlinear series $g_k^1$.\nSince the same property holds for an element\n$D'$ of the second $g_k^1$, the linear equivalence\n$nD\\sim nD'$ follows.\n\\end{proof}\n\n \n\\begin{example}\nLet $\\sigma$ be the order $k$ automorphism\nof ${\\mathbb P}^1\\times{\\mathbb P}^1$ defined by\n\\[\n \\sigma(x_0,x_1,y_0,y_1)=(\\zeta_k x_0,x_1, y_0,y_1),\n\\]\nwhere $\\zeta_k$ is a primitive $k$-th root of unity.\nWe now show that a curve $C\\in\\mathcal F_k$ \nwhich is $\\sigma$-invariant\nadmits an equation of the form\n\\[\n x_0^kg_2(y_0,y_1)+x_1^kf_2(y_0,y_1)=0,\n\\]\nwhere $f_2, g_2$ are homogeneous of degree \n$k$ in $y_0,y_1$. In particular the quotient\n$C\/\\langle\\sigma\\rangle$ \nhas genus zero. Thus  $C\\in\\mathcal F_k^{\\rm tor}$\nby either Proposition~\\ref{auto} or Proposition~\\ref{torsion}.\nThe automorphism $\\sigma$ preserves \neach ruling of the quadric and acts identically \non one of the two rulings.\nConsider a point in $\\mathcal H_1\\cap\\mathcal H_2$,\nwith the notation in the proof of Proposition~\\ref{torsion}, \nwhich corresponds to a $\\sigma$-invariant grid.\nThe lines of the grid which belong to the first ruling  \nare defined by either \n$x_0^k-x_1^k=0$ or $x_0^k=0$.\nAn equation of $C$ in such coordinates \nis then of the form\n\\[\n (x_0^k-\\mu x_1^k)h_1+(x_0^k+\\lambda x_1^k)h_2\n =\n x_0^k(h_1+h_2)+x_1^k(\\lambda h_2-\\mu h_1)\n =\n 0,\n\\]\nfor $\\mu\\in \\{0,1\\}$, $\\lambda\\in {\\mathbb C}$ and $h_1,h_2$ \nhomogeneous of degree three in $y_0,y_1$.\n\\end{example}\n\n\\begin{example}\n\\label{exa:new}\nThe moduli space of non-hyperelliptic \ncurves $C$ of genus four having an order five \nautomorphism $\\sigma$ such that \n$C\/\\langle\\sigma\\rangle$ \nhas genus zero is a 1-dimensional subvariety \nof $\\mathcal F_3^{\\rm tor}$.\nMoreover, any such $C$ is isomorphic to a \ncurve in the following family \n\\[\n x_0x_1^2y_1^3\n +\\alpha x_0^2x_1y_0^3\n +\\beta x_0^3y_0y_1^2\n +\\gamma x_1^3y_0^2y_1=0,\n\\]\nwhere $\\sigma(x_0,x_1,y_0,y_1)\n=(\\zeta_5x_0,x_1,\\zeta_5^3y_0,y_1)$ \nand $\\zeta_5$ is a primitive fifth root of unity.\nThe family contains curves which pass through the points \nof a grid of type $(5,5)$, for example the curve with\n\\[\n\\alpha=-\\zeta_5,\\ \\beta=\\zeta_5^3+\\zeta_5^2+\\zeta_5,\\ \\gamma=\\zeta_5^2+\\zeta_5. \n\\]\nHowever, the general element of the family is not \nof grilled type.\nThis means that if $D_i$ is a divisor of the $i$-th $g_3^1$ \nand $\\mathcal H_i\\subseteq |5D_1|\\cong{\\mathbb P}^{11}$ \nis the projectivization of the fifth symmetric \npower of $H^0(C,D_i)$, then the intersection\n$\\mathcal H_1\\cap\\mathcal H_2$ is empty.\nFor example this holds for the curve with $\\alpha=-1, \\beta=\\gamma=1$.\nThe statements for both curves can be checked \nby means of the Magma~\\cite{Magma} program\navailable here~\\url{http:\/\/www2.udec.cl\/~alaface\/software\/semiample\/aut}.\n\\end{example}\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{AC}{article}{\n   author={Arbarello, Enrico},\n   author={Cornalba, Maurizio},\n   title={Footnotes to a paper of Beniamino Segre: ``On the modules of\n   polygonal curves and on a complement to the Riemann existence theorem''\n   (Italian) [Math. Ann. {\\bf 100} (1928), 537--551;\\ Jbuch {\\bf 54}, 685]},\n   note={The number of $g^{1}_{d}$'s on a general $d$-gonal curve, and\n   the unirationality of the Hurwitz spaces of $4$-gonal and $5$-gonal\n   curves},\n   journal={Math. Ann.},\n   volume={256},\n   date={1981},\n   number={3},\n   pages={341--362},\n   issn={0025-5831},\n   review={\\MR{626954 (83d:14016)}},\n   doi={10.1007\/BF01679702},\n}\n\n\\bib{ADHL}{book}{\n    AUTHOR = {Arzhantsev, Ivan},\n    AUTHOR = {Derenthal, Ulrich},\n    AUTHOR = {Hausen, J\\\"urgen},\n    AUTHOR = {Laface, Antonio},\n    TITLE = {Cox rings},\n    series={Cambridge Studies in Advanced Mathematics},\n    volume={144},\n   publisher={Cambridge University Press, Cambridge},\n   date={2014},\n   pages={530},\n   isbn={9781107024625},\n}\n\n\n\\bib{BHPV}{book}{\n   author={Barth, Wolf P.},\n   author={Hulek, Klaus},\n   author={Peters, Chris A. M.},\n   author={Van de Ven, Antonius},\n   title={Compact complex surfaces},\n   series={Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A\n   Series of Modern Surveys in Mathematics [Results in Mathematics and\n   Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]},\n   volume={4},\n   edition={2},\n   publisher={Springer-Verlag, Berlin},\n   date={2004},\n   pages={xii+436},\n   isbn={3-540-00832-2},\n   review={\\MR{2030225 (2004m:14070)}},\n   doi={10.1007\/978-3-642-57739-0},\n}\n\n\\bib{Magma}{article}{\n    AUTHOR = {Bosma, Wieb},\n    AUTHOR = {Cannon, John},\n    AUTHOR = {Playoust, Catherine},\n     TITLE = {The {M}agma algebra system. {I}. {T}he user language},\n      NOTE = {Computational algebra and number theory (London, 1993)},\n   JOURNAL = {J. Symbolic Comput.},\n    VOLUME = {24},\n      YEAR = {1997},\n    NUMBER = {3-4},\n     PAGES = {235--265}\n}\n\n\n\\bib{CK}{article}{\n   author={Ciliberto, Ciro},\n   author={Kouvidakis, Alexis},\n   title={On the symmetric product of a curve with general moduli},\n   journal={Geom. Dedicata},\n   volume={78},\n   date={1999},\n   number={3},\n   pages={327--343},\n   issn={0046-5755},\n   review={\\MR{1725369 (2001e:14005)}},\n   doi={10.1023\/A:1005280023724},\n}\n\n\\bib{CP}{article}{\n   author={Collino, Alberto},\n   author={Pirola, Gian Pietro},\n   title={The Griffiths infinitesimal invariant for a curve in its Jacobian},\n   journal={Duke Math. J.},\n   volume={78},\n   date={1995},\n   number={1},\n   pages={59--88},\n   issn={0012-7094},\n   review={\\MR{1328752 (96f:14009)}},\n   doi={10.1215\/S0012-7094-95-07804-1},\n}\n\n\\bib{CPP}{article}{\n   author={Colombo, E.},\n   author={Pirola, G. P.},\n   author={Previato, E.},\n   title={Density of elliptic solitons},\n   journal={J. Reine Angew. Math.},\n   volume={451},\n   date={1994},\n   pages={161--169},\n   issn={0075-4102},\n   review={\\MR{1277298 (95e:58079)}},\n}\n\n\\bib{CS}{article}{\n   author={Cox, David},\n   author={Sidman, Jessica},\n   title={Secant varieties of toric varieties},\n   journal={J. Pure Appl. Algebra},\n   volume={209},\n   date={2007},\n   number={3},\n   pages={651--669},\n   issn={0022-4049},\n   review={\\MR{2298847 (2008i:14077)}},\n   doi={10.1016\/j.jpaa.2006.07.008},\n}\n\n\\bib{Gr}{article}{\n   author={Griffiths, Phillip A.},\n   title={Infinitesimal variations of Hodge structure. III. Determinantal\n   varieties and the infinitesimal invariant of normal functions},\n   journal={Compositio Math.},\n   volume={50},\n   date={1983},\n   number={2-3},\n   pages={267--324},\n   issn={0010-437X},\n   review={\\MR{720290 (86e:32026c)}},\n}\n\n\\bib{GH}{book}{\n   author={Griffiths, Phillip},\n   author={Harris, Joseph},\n   title={Principles of algebraic geometry},\n   series={Wiley Classics Library},\n   note={Reprint of the 1978 original},\n   publisher={John Wiley \\& Sons, Inc., New York},\n   date={1994},\n   pages={xiv+813},\n   isbn={0-471-05059-8},\n   review={\\MR{1288523 (95d:14001)}},\n   doi={10.1002\/9781118032527},\n}\n\n\\bib{Ke}{article}{\n   author={Keel, Se{\\'a}n},\n   title={Basepoint freeness for nef and big line bundles in positive\n   characteristic},\n   journal={Ann. of Math. (2)},\n   volume={149},\n   date={1999},\n   number={1},\n   pages={253--286},\n   issn={0003-486X},\n   review={\\MR{1680559 (2000j:14011)}},\n   doi={10.2307\/121025},\n}\n\n\\bib{K}{article}{  \n    AUTHOR = {Kond{\\=o}, Shigeyuki},\n     TITLE = {The moduli space of curves of genus 4 and {D}eligne-{M}ostow's\n              complex reflection groups},\n BOOKTITLE = {Algebraic geometry 2000, {A}zumino ({H}otaka)},\n    SERIES = {Adv. Stud. Pure Math.},\n    VOLUME = {36},\n     PAGES = {383--400},\n PUBLISHER = {Math. Soc. Japan},\n   ADDRESS = {Tokyo},\n      YEAR = {2002},\n   MRCLASS = {14H15 (14D07 14H45 14J28 32S40 33C80)},\n  MRNUMBER = {1971521 (2004h:14033)},\nMRREVIEWER = {I. Dolgachev},\n}\n\n\\bib{La}{book}{\n   author={Lazarsfeld, Robert},\n   title={Positivity in algebraic geometry. I},\n   series={Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A\n   Series of Modern Surveys in Mathematics [Results in Mathematics and\n   Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]},\n   volume={48},\n   note={Classical setting: line bundles and linear series},\n   publisher={Springer-Verlag, Berlin},\n   date={2004},\n   pages={xviii+387},\n   isbn={3-540-22533-1},\n   review={\\MR{2095471 (2005k:14001a)}},\n   doi={10.1007\/978-3-642-18808-4},\n}\n\n\\bib{R}{article}{\n   author={Raviolo, Emanuele},\n   title={A note on Griffiths infinitesimal invariant for curves},\n   journal={Ann. Mat. Pura Appl. (4)},\n   volume={193},\n   date={2014},\n   number={2},\n   pages={551--559},\n   issn={0373-3114},\n   review={\\MR{3180933}},\n   doi={10.1007\/s10231-012-0290-x},\n}\n\n\n \\end{biblist}\n \\end{bibdiv}\n\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\\IEEEPARstart{W}{e}  study the problem of extracting a prescribed number of random bits by reading the smallest possible number of symbols from imperfect stochastic processes. For perfect stochastic processes, including processes with known accurate distributions or perfect biased coins, this problem has been well studied. It dates back to von Neumann [9] who considered the problem of generating random bits from a biased coin with unknown probability. Recently, in \\cite{Zhou12_Streaming},  we improved von Neumann's scheme and introduced an algorithm that generates `random bit streams' from biased coins, uses bounded space and runs in expected linear time. This algorithm can generate a prescribed number of random bits with an asymptotically optimal efficiency. On the other hand, efficient algorithms have also been developed for extracting randomness from any known stochastic process (whose distribution is given). In \\cite{Knuth1976}, Knuth and Yao presented a simple procedure for generating sequences with\narbitrary probability distributions from an unbiased coin (the probability of H and T is $\\frac{1}{2}$). In \\cite{Abrahams1996}, Abrahams considered\na source of biased coin whose distribution is an integer power of a noninteger.\nHan and Hoshi \\cite{Han1997}  studied the general problem and proposed an interval algorithm that generates a prescribed number of random bits from any known stochastic process and achieves the information-theoretic upper bound on efficiency.  However, in practice, sources of stochastic processes have inherent correlations and are affected by measurement's noise, hence, they are not perfect. Existing algorithms for extracting randomness from perfect stochastic processes cannot work for imperfect stochastic processes, where uncertainty exists.\n\nTo extract randomness from an imperfect stochastic process, one approach is to apply a seeded or seedless extractor to a sequence generated by the process that contains a sufficient amount of  randomness, and we call this approach as a fixed-length extractor for stochastic processes since all the possible input sequences have the same fixed length. Efficient constructions of seeded or seedless extractors have been extensively studied in last two decades, and it shows that the number of random bits extracted by them can approach the source's min-entropy asymptotically \\cite{Dvir08, Nis96, Sha02,Rao2007,Kamp11}. Although fixed-length extractors can generate random bits with good quality from imperfect stochastic processes,\ntheir efficiencies are not close to the optimality. Here, we define the \\emph{efficiency} of an extractor for stochastic processes as the asymptotic ratio\nbetween the number of extracted random bits and the entropy of its input sequence (the entropy of its input sequence is proportional to the expected input length if the stochastic process is stationary ergodic), which is upper bounded by $1$ since the process of extracting randomness does not increase entropy. Based on this definition, we can conclude that the efficiency of a fixed-length extractor is upper bounded by the ratio between the min-entropy and the entropy of the input sequence, which is usually several times smaller than $1$. So fixed-length extractors are not very efficient in extracting randomness from stochastic processes. The intuition is that, in order to minimize the expected number of symbols read from an imperfect stochastic process, the length of the input sequence should be adaptive, not being fixed.\n\nThe concept of min-entropy and entropy are defined as follows.\n\\begin{Definition}\nGiven a random source $X$ on $\\{0,1\\}^n$, the \\emph{min-entropy} of $X$ is defined as\n$$H_{\\min}(X)=\\min_{x\\in \\{0,1\\}^n } \\log \\frac{1}{P[X=x]}.$$\nThe \\emph{entropy} of $X$ is defined as\n$$H(X)=\\sum_{x\\in \\{0,1\\}^n } P[X=x] \\log \\frac{1}{P[X=x]}.$$\n\\end{Definition}\n\nThe following example is constructed for comparing entropy with min-entropy for a simple random variable.\n\\begin{Example}\nLet $X$ be a random variable such that $P[X=0]=0.9$ and $P[X=1]=0.1$, then\n$H_{\\min}(X)=0.152$ and $H(X)=0.469$. In this case, the entropy of $X$ is about three times its min-entropy.\\hfill\n\t$\\Box$\n\\end{Example}\n\nIn this paper, we focus on the notion and constructions of variable-length extractors (short for variable-to-fixed length extractors), namely, extractors with variable input length and fixed output length. (Note that the interval algorithm proposed by Han and Hoshi \\cite{Han1997} and the streaming algorithm proposed by us \\cite{Zhou12_Streaming} are special cases of variable-length extractors). Our goal is to extract a prescribed number of random bits in the sense of statistical distance while minimizing the expected input cost, measured by the entropy of the input sequence (whose length is variable). To make this precise, we let\n$d(\\mathcal{R},\\mathcal{M})$ be the difference between two known stochastic processes $\\mathcal{R}$ and $\\mathcal{M}$, defined by\n$$d(\\mathcal{R},\\mathcal{M})=\\limsup_{n\\rightarrow\\infty} \\max_{x\\in \\{0,1\\}^n} \\frac{\\log_2\\frac{P_{\\mathcal{R}}(x)}{P_{\\mathcal{M}}(x)}}{\\log_2\\frac{1}{P_{\\mathcal{M}}(x)}},$$\nwhere $P_{\\mathcal{R}}(x)$ is the probability of generating $x$ from $\\mathcal{R}$ when the sequence length is $|x|$, and $P_{\\mathcal{M}}(x)$ is the probability of generating $x$ from $\\mathcal{M}$ when the sequence length is $|x|$.\n\nA few models of imperfect stochastic processes are introduced and investigated, including,\n\\begin{itemize}\n  \\item Let $\\mathcal{M}$ be a known stochastic process, we consider an arbitrary stochastic process $\\mathcal{R}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a constant $\\beta$.\n  \\item We consider $\\mathcal{R}$ as an arbitrary stochastic process such that\n  $\\min_{\\mathcal{M}\\in \\mathcal{G}_{s.e.}}d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for\n  a constant $\\beta$, where $\\mathcal{G}_{s.e.}$ denotes the set consisting of all stationary ergodic processes.\n\\end{itemize}\n\nGenerally, given a real slight-unpredictable source $\\mathcal{R}$, it is not easy to estimate the exact value of $d(\\mathcal{R},\\mathcal{M})$ for\na stochastic process $M$. But its upper bound, i.e., $\\beta$, can be easily obtained.\nThe parameter\n$\\beta$ describes how unpredictable the real source $\\mathcal{R}$ is, so we call it the \\emph{uncertainty} of $\\mathcal{R}$.\nWe prove that it is impossible to construct an extractor that achieves efficiency strictly larger than $1-\\beta$\nfor all the possible sources $\\mathcal{R}$ with uncertainty $\\beta$. Then we introduce several constructions of variable-length extractors, and show that their efficiencies can reach $\\eta \\geq 1-\\beta$; that is, the constructions are asymptotically optimal. The proposed variable-length extractors have two benefits: (i) they are generalizations of algorithms for perfect sources to  address general imperfect sources; and (ii) they bridge the gap between min-entropy and entropy on efficiency.\n\nThe following example is constructed to compare the performances of a variable-length extractor and\na fixed-length extractor when extracting randomness from a slightly-unpredictable independent process.\n\\begin{Example} Consider an independent process $x_1x_2x_3...$ such that $P[x_i=1]\\in [0.9, 0.91]$, then it can be obtained that $\\beta\\leq 0.0315$.\nFor this source, a variable-length extractor can generate random bits with efficiency at least $1-\\beta=0.9685$ that is very close to the upper bound $1$. In comparison, fixed-length extractors can only reach the efficiency at most $0.3117$.\n\\end{Example}\n\n\nThe remainder of this paper is organized as follows. Section \\ref{sec_pre} presents background and related results.\nIn Section \\ref{var_sec_efficiency}, we demonstrate that one cannot construct a variable-length extractor with efficiency strictly larger than $1-\\beta$ when the source has uncertainty $\\beta$. Then we focus on the seeded constructions of variable-length extractors, namely, we use a small number of additional truly random bits as the seed (catalyst).\nThree different constructions are provided and analyzed in Section \\ref{var_section_tech1}, Section \\ref{var_section_tech2} and Section \\ref{var_section_tech4}\nseparately. All these constructions have efficiencies lower bounded by $1-\\beta$, implying their optimality.\nFinally, we discuss seedless constructions of variable-length extractors for some types of random sources in Section \\ref{sec_randomnessextraction}, followed by the concluding remarks.\n\n\n\\section{Preliminaries}\n\\label{sec_pre}\n\n\\subsection{Statistical Distance}\n\n\\emph{Statistical Distance} is used in computer science to measure the difference between two distributions. Let $X$ and $Y$ be two random sequences with range $\\{0,1\\}^m$, then the statistical distance between $X$ and $Y$ is defined as\n$$\\|X-Y\\|=\\max_{T:\\{0,1\\}^m\\rightarrow \\{0,1\\}} |P[T(X)=1]-P[T(Y)=1]|$$\nover a boolean function $T$.\nWe say that $X$ and $Y$ are $\\epsilon$-close if $\\|X-Y\\|\\leq \\epsilon$. According to this definition, we can also write\n$$\\|X-Y\\|=\\frac{1}{2}\\sum_{x\\in \\{0,1\\}^m}|P[X=x]-P[Y=x]|\\leq \\epsilon.$$\nIt is equivalent to the former expression.\n\nLet $U_m$ denote the uniform distribution on $\\{0,1\\}^m$. In order to let a sequence $Y$ to be able to take place of the truly random bits in a randomized application, we let $Y$ be $\\epsilon$-close to $U_m$, where $\\epsilon$ is small enough. In this case,\nthe extra probability error introduced by this replacement is at most $\\epsilon$.\nIn this paper, we want to extract $m$ almost-random bits such that they form a sequence $\\epsilon$-close to the uniform distribution $U_m$ on $\\{0,1\\}^m$ with specified small $\\epsilon>0$, i.e.,\n$$\\|Y-U_m\\|\\leq \\epsilon.$$\n\n\\subsection{Seeded Extractors}\n\nIn 1990, Zuckerman introduced a general model of weak random sources, called $k$-sources, namely whose min-entropy is at least $k$ \\cite{Zuc90}.\nIt was shown that given a source on $\\{0,1\\}^n$ with min-entropy $k<n$, it is impossible to devise a single function that extracts even one bit of randomness. This observation led to the introduction of seeded extractors, which use a small number of additional truly random bits as the seed (catalyst). When simulating a probabilistic algorithm, one can simply eliminate the requirement of truly random bits by enumerating all possible strings for the seed and taking a majority vote on the final results. There are a variety of very efficient constructions of seeded extractors, summarized in \\cite{Dvir08, Nis96, Sha02}. Mathematically, a seeded extractor\nis a function,\n$$E: \\{0,1\\}^n \\times \\{0,1\\}^d \\rightarrow \\{0,1\\}^m,$$\nsuch that for every distribution $X$ on $\\{0,1\\}^n$ with $H_{\\min}(X)\\geq k$, the distribution $E(X,U_d)$ is $\\epsilon$-close to\nthe uniform distribution $U_m$. Here, $d$ is the seed length, and we call such an extractor as a $(k,\\epsilon)$ extractor.\nThere are a lot of works focusing on efficient constructions of seeded extractors. A standard application of the probabilistic method \\cite{Radhakrishnan2000} shows that there exists a seeded extractor which can extract asymptotically $H_{\\min}(X)$ random bits with $\\log(n-H_{\\min}(X))$ additional truly random bits. Recently, Guruswami, Umans and Vadhan \\cite{Guruswami07} provided an explicit construction of seeded extractors, whose efficiency is  very close to the bound obtained based on the probabilistic method. Their main result is described as follows:\n\n\\begin{Lemma} \\emph{\\cite{Guruswami07}} \\label{lemma_seededextractor} For every constant $\\alpha>0$, and all positive integers $n, k$ and all $\\epsilon>0$, there is an explicit construction of a $(k,\\epsilon)$ extractor $E: \\{0,1\\}^n\\times \\{0,1\\}^d\\rightarrow\\{0,1\\}^m$ with $d\\leq \\log n+ O(\\log(k\/\\epsilon))$ and $m\\geq (1-\\alpha)k$.\n\\end{Lemma}\n\nThe above result implies that given any source $X \\in \\{0,1\\}^n$ with min-entropy $k$, if $\\geq (1+\\alpha)m$ with $\\alpha>0$, we can always construct\na seeded extractor to generates a random sequence $Y\\in \\{0,1\\}^m$ that is $\\epsilon$-close to the uniform distribution. In this case,\nthe seed length $d\\leq \\log n + O(\\log(k\/\\epsilon))$ depends on the input length $n$ and the parameter $\\epsilon$.\n\n\\subsection{Seedless Extractors}\n\nIn the last decade, the concept of seedless (deterministic) extractors has attracted renewed interests, motivated by the reduction of the computational complexity for simulating probabilistic algorithms as well as some requirements in cryptography \\cite{Dodis00}. Several specific classes of sources have been studied, including independent sources, which can be divided into several independent parts containing certain amount of randomness \\cite{Barak06, Rao2007, Raz2005}; bit-fixing sources, where some bits in a binary sequence are truly random and the remaining bits are fixed \\cite{Cohen89, Gabizon06, Kamp06}; samplable sources, where the source is generated by a process that has a bounded amount of computational resources like space \\cite{Kamp11, Trevisan00}. For example, suppose that we have multiple independent sources with the same length $n$. It is known how to extract from two sources when\nthe min-entropy in each is $\\geq 0.5n$ \\cite{Raz2005} or slightly less than $0.5n$ \\cite{Bou05}, how to extract from $O(1\/\\gamma)$ sources if the min-entropy in each is at least $n^\\gamma$ \\cite{Rao06}. All these constructions have exponentially small error, and they are able to extract $\\Theta(k)$ random bits.\n\nBoth seeded extractors and seedless extractors described above have fixed input length, fixed seed length ($d=0$ for seedless extractors) and fixed output length. So we call them fixed-length extractors. To apply fixed-length extractors in extracting randomness from a stochastic process, it needs to first read a sequence of fixed length, whose min-entropy is strictly larger than the number of random bits that we need to generate. Fixed-length extractors can generate random bits of good quality from imperfect stochastic processes,\nbut they usually consume more incoming symbols than what are necessarily required. To increase information efficiency,\nwe let the length of input sequences be adaptive, hence, we have the concept of `variable-length extractors'.\n\n\\subsection{Variable-Length Extractors}\n\nA variable-length extractor is an extractor with variable input length and fixed output length. When applying a variable-length extractor to\na stochastic process, it reads incoming symbols one by one until the whole incoming sequence meets certain criterion, then it maps the incoming sequence into a binary sequence of fixed length as the output. Depending on the sources, the construction may require a small number of additional truly random bits as the seed. Hence, we have seeded variable-length extractors and seedless variable-length extractors.\n\nA seeded variable-length extractor is a function,\n$$V_E: S_p\\times\\{0,1\\}^d\\rightarrow\\{0,1\\}^m,$$\nsuch that given a real source $\\mathcal{R}$, the output sequence is $\\epsilon$-close to\nthe uniform distribution $U_m$. Here, $S_p$ is the set consisting of all possible input sequences, called the input set. It is complete and prefix-free. The input sequence is compete, that means, any infinite sequence has a prefix in the set; so when reading symbols from any source, we can always meet a sequence in the set. Then we stop reading and map this sequence into a binary sequence of length $m$. Being prefix-free is not very necessary; it ensures that all the sequences in $S_p$ are possible to read.\n\nA general procedure of extracting randomness by using variable-length extractors can be divided into three steps:\n\n\\begin{enumerate}\n  \\item Determining an input set $S_p$ such that its min-entropy based on the real source $\\mathcal{R}$ is at least $k$, namely,\n$$\\min_{x\\in S_p}\\log_2\\frac{1}{P_\\mathcal{R}(x)}\\leq k,$$\nwhere $k\\geq (1+\\alpha)m$ for any $\\alpha>0$.\n  \\item We construct an injective function\n$$V:S_p\\rightarrow \\{0,1\\}^n,$$\nto map the sequences in $S_p$ into binary sequences of length $m$. We read symbols from the source $\\mathcal{R}$ one by one until\nthe current incoming sequence matches one in $S_p$. This incoming sequence is then mapped to a binary sequence of length $n$ based on function $V$. As a result, we get a random sequence $Z$ with length $n$ and min-entropy $k$ (since $V$ is injective).\n  \\item Since $k=(1+\\alpha)$ with an $\\alpha>0$, according to Lemma \\ref{lemma_seededextractor}, we can always find a seeded extractor,\n  $$E:\\{0,1\\}^n \\times\\{0,1\\}^d\\rightarrow\\{0,1\\}^m$$\n  that can extract $m$ almost-random bits from a source with min-entropy $k$. By applying  this seeded extractor $E$ to the sequence $Z$,\n  we get a random sequence of length $m$ that is $\\epsilon$-close to the uniform distribution $U_m$. Here, the seed length $d\\leq \\log n + O(\\log(k\/\\epsilon))$.\n\\end{enumerate}\n\nWe can see that the construction of a variable-length extractor is a cascade of\na function $V$ and a seeded extractor $E$, i.e.,\n$$V_E=E\\bigotimes V.$$\n\nNote that our requirement is to extract a sequence of $m$ almost-random bits that is $\\epsilon$-close to the uniform distribution $U_m$. The key of constructing variable-length extractors is to find the input set $S_p$ with min-entropy $k$, even the distribution of the real source $\\mathcal{R}$ is slightly unpredictable, such that the expected length of the sequences in $S_p$ is minimized. For stationary ergodic processes, minimizing the expected length is equivalent to minimizing the entropy of the sequences in $S_p$ asymptotically (this will be discussed in this section).\n\nFor some specific types of sources, including independent sources and samplable sources, by applying the ideas in \\cite{Rao2007} and \\cite{Kamp11} we can remove\nthe requirement of truly random bits without degrading the asymptotic performance. As a result, we have seedless variable-length extractors.\nFor example, if the source $\\mathcal{R}$ is an independent process, we can first apply the method in \\cite{Rao2007} to extract $d$ almost-random bits from the first $\\Theta(\\log \\frac{m}{\\epsilon})$ bits, and then use them as the seed of a seeded variable-length extractor to extract randomness from the rest of the process. The detailed discussions will be given in Section \\ref{sec_randomnessextraction}.\n\n\\section{Efficiency and Uncertainty}\n\\label{var_sec_efficiency}\n\n\\subsection{Efficiency}\n\nTo consider the performance of an extractor, we define its \\emph{efficiency} as\nthe asymptotical ratio between the output length and the total entropy of all its inputs. So the efficiency of an extractor can be written as\n$$\\eta=\\lim_{m\\rightarrow\\infty} \\frac{m}{H_{\\mathcal{R}}(X_m)+d},$$\nsuch that the output sequence is $\\epsilon$-close to the uniform distribution $U_m$\non $\\{0,1\\}^m$, where $\\epsilon$ is small, $d$ is the seed length, $m$ is the output length, and $H_{\\mathcal{R}}(X_m)$ is the entropy of the input sequence $X_m$ with range on $S_p$.\nIn our constructions,  $d\\leq \\log n + O(\\log(m\/\\epsilon))$, which is ignorable compared to $H_{\\mathcal{R}}(X_m)$ when $m\\rightarrow\\infty$. Hence,\nwe can write\n$$\\eta=\\lim_{m\\rightarrow\\infty} \\frac{m}{H_{\\mathcal{R}}(X_m)}.$$\nIn the definition, we use the entropy of the input sequence rather than the expected input length, because the source that we considered may not be stationary ergodic.\nIt needs to mention that, in seeded constructions, the value of $d$ is also an important parameter although it is much smaller than $m$. The problem of minimizing the seed length $d$ can be studied separately from minimizing the entropy of the input sequence, and it will be addressed in this paper.\n\nFirst, we demonstrate that if a distribution is $\\epsilon$-close to the uniform distribution $U_m$, then\nthe entropy of this distribution is asymptotically $m$ for any $\\epsilon<1$.\n\n\\begin{Lemma}\\label{var_lemma2}\nLet $X$ be a random sequence on $\\{0,1\\}^m$ that is $\\epsilon$-close to the uniform distribution $U_m$, then\n$$ m- \\log_2\\frac{1}{1-\\epsilon}\\leq H(X)\\leq m.$$\n\\end{Lemma}\n\n\\proof Since there are totally $2^m$ possible assignments for $X$, it is easy to get $H(X)\\leq m$.\nSo we only need to prove that\n$$H(X)\\geq m- \\log_2\\frac{1}{1-\\epsilon}.$$\n\nLet $p(x)$ denote $P[X=x]$ for $x\\in \\{0,1\\}^m$.\nSince $X$ is  $\\epsilon$-close to the uniform distribution $U_m$, we have\n$$\\frac{1}{2} \\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|\\leq \\epsilon.$$\n\nThen the lower bound of $H(X)$ can be written as\n$$\\min_{p} \\sum_{x\\in \\{0,1\\}^m} p(x) \\log_2\\frac{1}{p(x)}$$\nsubject to\n$$p(x)\\geq 0, \\forall x\\in \\{0,1\\}^m;$$\n$$\\sum_{x\\in\\{0,1\\}^m}p(x)=1;$$\n$$\\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|\\leq 2\\epsilon.$$\n\nObviously, the optimal solution of the above  problem happens at\n$$\\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|=2\\epsilon.$$\n\nTo solve the problem based on Lagrange Multipliers, we let\n$$\\lambda(p)= \\sum_{x\\in \\{0,1\\}^m}\\ p(x) \\log_2\\frac{1}{p(x)}+\\lambda_1(\\sum_{x\\in\\{0,1\\}^m}p(x)-1)$$\n$$+\\lambda_2 (\\sum_{x\\in \\{0,1\\}^m} \\|p(x)-2^{-m}\\|-2\\epsilon).$$\n\nIf $p(x)\\geq 0$ with $x\\in\\{0,1\\}^m$ is a solution of the above question, then\n$$\\frac{\\partial \\lambda }{\\partial(p(x))}=0,$$\ni.e.,\n$$ \\left\\{ \\begin{array}{cc}\n          \\frac{\\ln p(x)+1}{\\ln 2}+ \\lambda_1+\\lambda_2=0    &  \\textrm{ if } 2^{-m}\\leq p(x)\n          \\leq 1, \\\\\n           \\frac{\\ln p(x)+1}{\\ln 2}+ \\lambda_1-\\lambda_2=0   &  \\textrm{ if } 0\\leq p(x)\\leq 2^{-m}.\n           \\end{array}\n\\right.$$\n\nSo there exists two constants $a$ and $b$ with $0\\leq a\\leq 2^{-m}\\leq b\\leq 1$, such that,\n$$ \\left\\{ \\begin{array}{cc}\n          p(x)=a   &  \\textrm{ if } 2^{-m}\\leq p(x)\\leq 1, \\\\\n          p(x)=b   &  \\textrm{ if } 0\\leq p(x)\\leq 2^{-m}.\n           \\end{array}\n\\right.$$\n\nAssume that there are $t$ assignments of $x$ with $p(x)=a$, then there are $2^m-t$ assignments of $x$ with $p(x)=b$.\nHence, the problem is converted to the one over $a,b,t$, i.e.,\n$$\\min_{a,b,t} t a\\log \\frac{1}{a} +(2^m-t) b\\log \\frac{1}{b},$$\nsubject to\n$$0\\leq t\\leq 2^m;$$\n\\begin{equation}ta +(2^m-t)b=1;\\label{var_equ_entropy1}\\end{equation}\n\\begin{equation}t(2^{-m}-a)+(2^m-t)(b-2^{-m})=2\\epsilon.\\label{var_equ_entropy2}\\end{equation}\nFrom Equ.~(\\ref{var_equ_entropy1}) and (\\ref{var_equ_entropy2}), we get\n$$a=2^{-m}-\\frac{\\epsilon}{t}, \\quad b=2^{-m}+\\frac{\\epsilon}{2^m-t}.$$\n\nSo the question is finding the optimal $t$ that minimizes\n$$-t(2^{-m}-\\frac{\\epsilon}{t})\\log_2 (2^{-m}-\\frac{\\epsilon}{t})$$\n$$-(2^m-t)(2^{-m}+\\frac{\\epsilon}{2^m-t})\\log_2(2^{-m}+\\frac{\\epsilon}{2^m-t}),$$\nsubject to\n$$0\\leq t\\leq \\frac{\\epsilon}{2^{-m}}.$$\n\nThe optimal solution is $t^*=\\frac{\\epsilon}{2^{-m}}$. In this case, the entropy of $X$ is\n$$H(X)= \\log (2^m-t)=m-\\log_2\\frac{1}{1-\\epsilon},$$\nwhich is the lower bound.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nIn the following lemma, we show that for any extractor, its efficiency is upper bounded by $1$. The reason is that the amount of information, i.e., entropy, does not increase\nduring the process of randomness extraction.\n\n\\begin{Lemma}\\label{lemma_efficiency1}\nFor any extractor with seed length $d$ and output length $m$, if $d=o(m)$, its efficiency $\\eta\\leq 1$.\n\\end{Lemma}\n\n\\proof We consider fixed-length extractors as a special case of variable-length extractors, and consider seedless extractors as a special case of seeded extractors when $d=0$. So  our proof only focus on seeded variable-length extractors.\n\nA main observation is that for any extractor, the entropy of its output sequence is bounded\nby the entropy of the input sequence plus the entropy of the seed, since the process of extracting randomness cannot create new randomness.\n\nFor the output sequence, denoted by $Y$, it is $\\epsilon$-close to the uniform distribution $U_m$. According to\nLemma \\ref{var_lemma2}, its entropy is $$H_{\\mathcal{R}}(Y)\\geq m-\\log_2\\frac{1}{1-\\epsilon}.$$\n\nThe total entropy of the inputs is $H_{\\mathcal{R}}(X_m)+d$. Hence,\n$$H_{\\mathcal{R}}(Y)\\leq H_{\\mathcal{R}}(X_m)+d.$$\n\nAs a result, the efficiency of the extractor is\n$$\\eta=\\lim_{m\\rightarrow\\infty} \\frac{m}{H_{\\mathcal{R}}(X_m)}=\\lim_{m\\rightarrow\\infty}\\frac{H_{\\mathcal{R}}(Y)}{H_{\\mathcal{R}}(X_m)+d}\\leq 1.$$\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nIf $\\mathcal{R}$ is a stationary ergodic process, we define its entropy rate as\n$$h(\\mathcal{R})=\\lim_{l\\rightarrow\\infty}\\frac{H(X^l)}{l},$$\nwhere $X^l$ is a random sequence of length $l$ generated from the source $\\mathcal{R}$.\nIn this case, the entropy of the input sequence on $S_p$ is proportional to the expected input length.\n\n\\begin{Lemma} \\label{var_theorem_stionaryergodiclength} Given a stationary ergodic source $\\mathcal{R}$, let $X_m$ be the input sequence of a variable-length extractor that has an output length $m$. Then\n$$\\lim_{m\\rightarrow\\infty} \\frac{H_{\\mathcal{R}}(X_m)}{E_{\\mathcal{R}}[|X_m|]}=h(\\mathcal{R}),$$\nwhere $E_{\\mathcal{R}}[|X_m|]$ is the expected input length.\n\\end{Lemma}\n\n\\proof $X_m$ is a random sequence from $S_p$ based on the distribution of $\\mathcal{R}$.\nLet $l_1$ be the minimum length of the sequences in $S_p$, as $m\\rightarrow\\infty$, $l_1\\rightarrow\\infty$.\nNow, we define $$l_i=l_1+(i-1)\\log l_1 \\textrm{ for all }i\\geq 1.$$ Based on them, we divide all the sequences in $S_p$ into subsets\n$$S_i=\\{x|x\\in S_p, l_i\\leq |x|\\leq l_{i+1}-1\\}$$\nfor $i\\geq 1$.\n\nLet $p_i=P_{\\mathcal{R}}(X_m\\in S_i)$, then\n$$H_{\\mathcal{R}}(X_m)\n   \\geq \\sum_{i}[(\\sum_{j>i} p_j)H_{\\mathcal{R}}(X_{l_{i-1}+1}^{l_i}|X_{1}^{l_{i-1}}, |X_m|\\geq l_i)],$$\nwhere $l_0=0$, $\\sum_{j>i} p_j$ is the probability that $|X_m|\\geq l_i$, and\n$X_a^b$ is a sequence of $X_m$ from the $a$th element to the $b$th element.\n\nSince $X_m$ is generated from a stationary ergodic process, and $l_i-{l_{i-1}}\\rightarrow\\infty$ as $m\\rightarrow\\infty$, we can get\n$$H_{\\mathcal{R}}(X_{l_{i-1}+1}^{l_i}|X_{1}^{l_{i-1}},  |X_m|\\geq l_i)\\rightarrow (l_i-l_{i-1})h(\\mathcal{R}).$$\n\nAs a result, as $l_1\\rightarrow\\infty$, we have\n\\begin{eqnarray*}\nH_{\\mathcal{R}}(X_m)&\\geq& (1-\\epsilon) \\sum_{i}(\\sum_{j>i} p_j) (l_i-l_{i-1})h(\\mathcal{R})\\\\\n&=&(1-\\epsilon)\\sum_i p_i l_i h(\\mathcal{R}),\n\\end{eqnarray*}\nfor an arbitrary $\\epsilon>0$.\n\nAlso considering the other direction, we can get\nthat as $l_1\\rightarrow\\infty$,\n\\begin{eqnarray*}\n H_{\\mathcal{R}}(X_m)\n&\\leq &(1+\\epsilon)\\sum_i p_i l_{i+1} h(\\mathcal{R})\\\\\n&=& (1+\\epsilon)\\sum_i p_i (l_i+\\log l_1) h(\\mathcal{R}),\n\\end{eqnarray*}\nfor an arbitrary $\\epsilon>0$.\n\nFor the expected input length, i.e., $E_{\\mathcal{R}}[|X_m|]$, it is easy to show that\n$$\\sum_i p_i l_i\\leq E_{\\mathcal{R}}[|X_m| ] \\leq \\sum_i p_i l_{i+1}=\\sum_i p_i( l_i+\\log l_1)  .$$\n\nSo as $m\\rightarrow \\infty$, i.e., $l_1\\rightarrow\\infty$, it yields\n$$\\lim_{m\\rightarrow\\infty} \\frac{H_{\\mathcal{R}}(X_m)}{E_\\mathcal{R}[|X_m|]}=\\lim_{m\\rightarrow\\infty}\\frac{\\sum_i p_i l_i h(\\mathcal{R})}{\\sum_i p_i l_i}$$\n$$=h(\\mathcal{R}).$$\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\n\\subsection{Sources and Uncertainty}\n\nGiven a source $\\mathcal{R}$, if its distribution is known, we say that this source is a known stochastic process, and its uncertainty is $0$. In this paper,\nwe mainly focus on those imperfect processes whose distributions are slightly unpredictable due to many factors like\nthe existence of external adversaries.\n\nFirst, given two known stochastic processes $\\mathcal{R}$ and $\\mathcal{M}$, we let\n$d(\\mathcal{R},\\mathcal{M})$ be the difference between $\\mathcal{R}$ and $\\mathcal{M}$.  Here, we define $d(\\mathcal{R},\\mathcal{M})$ as\n$$d(\\mathcal{R},\\mathcal{M})=\\limsup_{n\\rightarrow\\infty} \\max_{x\\in \\{0,1\\}^n} \\frac{\\log_2\\frac{P_{\\mathcal{R}}(x)}{P_{\\mathcal{M}}(x)}}{\\log_2\\frac{1}{P_{\\mathcal{M}}(x)}},$$\nwhere $P_{\\mathcal{R}}(x)$ is the probability of generating $x$ from $\\mathcal{R}$ when the sequence length is $|x|$, and $P_{\\mathcal{M}}(x)$ is the probability of generating $x$ from $\\mathcal{M}$ when the sequence length is $|x|$. Although there are some existing ways such as normalized Kullback-Leibler divergence to measure the difference between two sources, with them it is not easy to estimate the uncertainty of a source and it is\nnot easy to analyze the performances of constructed variable-length extractors.\n\nIn the rest of this paper, we investigate a few models of unpredictable sources. Most natural source can be well described in those ways.\n\n\\begin{enumerate}\n  \\item The source $\\mathcal{R}$ is an arbitrary stochastic process such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a constant $\\beta\\in [0,1]$ and\n  a known stochastic process $\\mathcal{M}$.\n  \\item $\\mathcal{R}$ is an arbitrary stochastic process such that there exists a stationary ergodic process $\\mathcal{M}$ (whose distribution is unknown)\nand $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$; that is, $\\min_{\\mathcal{M}\\in \\mathcal{G}_{s.e.}}d(\\mathcal{R},\\mathcal{M})\\leq \\beta$,  where $\\mathcal{G}_{s.e.}$ denotes the set consisting of all stationary ergodic processes.\n\\end{enumerate}\n\nIn both the models, we call $\\beta$ as the \\emph{uncertainty} of the source $\\mathcal{R}$. In the real world,\n$\\beta$ can be easily estimated without knowing the distribution of the processes.\nIt just reflects how unpredictable the real source $\\mathcal{R}$ is.\n\nTo construct variable-length extractors, we only care about the possible input sequences, namely, those in $S_p$. Hence, for the case\nof finite length, $d_p(\\mathcal{R},\\mathcal{M})$ is a more important parameter for us, defined by\n$$d_p(\\mathcal{R},\\mathcal{M})= \\max_{x\\in S_p} \\frac{\\log_2\\frac{P_{\\mathcal{R}}(x)}{P_{\\mathcal{M}}(x)}}{\\log_2\\frac{1}{P_{\\mathcal{M}}(x)}},$$\n\nAs the number of required random bits $m$ increases, $d_p(\\mathcal{R},\\mathcal{M})$ quickly converge to $d(\\mathcal{R}, \\mathcal{M})$.\nAnd we can write\n$$d_p(\\mathcal{R},\\mathcal{M})=d(\\mathcal{R}, \\mathcal{M})+\\epsilon_p$$\nfor a very small constant $\\epsilon_p$. As $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. In this case, the upper bound of $d_p(\\mathcal{R},\\mathcal{M})$ or\n$\\min_{\\mathcal{M}\\in \\mathcal{G}_{s.e.}}d_p(\\mathcal{R},\\mathcal{M})$ is\n$$\\beta_p=\\beta+\\epsilon_p.$$\n\n\\begin{Example} Let $x_1x_2...\\in \\{0,1\\}^*$ be a sequence generated from an independent source $\\mathcal{R}$ such that $$\\forall i\\geq 1, P[x_i=1]\\in [0.8,0.82].$$\nIf we let $\\mathcal{M}$ be a biased coin with probability $0.8132$, then\n$$\\beta=\\max_{\\textrm{possible }\\mathcal{R}} d(\\mathcal{R},\\mathcal{M})$$\n$$=\\max(\\frac{\\log_2 \\frac{0.2}{0.1868}}{\\log_2 \\frac{1}{0.1868}}, \\frac{\\log_2 \\frac{0.82}{0.8132}}{\\log_2 \\frac{1}{0.8132}})=0.0405.$$ \\hfill  $\\Box$\n\\end{Example}\n\nAccording to our definition, $d(\\mathcal{M},\\mathcal{R})\\leq \\beta$ if and only if\n$$P_{\\mathcal{R}}(x)\\leq P_{\\mathcal{M}}(x)^{1-\\beta}$$\nfor all $x\\in\\{0,1\\}^\\infty$ with  $|x|\\rightarrow\\infty$. This is a condition that is very easy to be satisfied by many natural stochastic processes for a small $\\beta$.\n\n\\begin{Lemma}  If $d(\\mathcal{R},\\mathcal{M})\\rightarrow 0$, we have\n$$P_{\\mathcal{R}}(x)\\rightarrow P_{\\mathcal{M}}(x) $$\nfor all $x\\in \\{0,1\\}^*$.\n\\end{Lemma}\n\n\\subsection{Efficiency and Uncertainty}\n\nIn this subsection, we investigate the relation between the efficiency and uncertainty. We show that given\na stochastic process $\\mathcal{R}$ with uncertainty $\\beta$, as described in the previous subsection, one cannot construct a variable-length extractor with efficiency strictly larger than $1-\\beta$ for all the possibilities of $\\mathcal{R}$.\n\nLet us first consider a simple example: let $X$ be a random sequence with the uniform distribution on $\\{0,1\\}^n$ and let $Y$\nbe an arbitrary random sequence on $\\{0,1\\}^n$ such that $$\\frac{\\log_2 \\frac{ P[Y=x]}{P[X=x]}}{\\log_2\\frac{1}{P[X=x]}}\\leq \\beta, \\forall x\\in \\{0,1\\}^n.$$\nNow, we show that from the source $Y$, one cannot construct an extractor with efficiency strictly larger than $1-\\beta$.\nTo see this, we consider an extractor $f$ with output length $m$, and a source $Y$ with\n$$P[Y=y]\\in \\{0, 2^{-n(1-\\beta)}\\}, \\forall y\\in \\{0,1\\}^n.$$\nFor this a source $Y$, its entropy is $H(Y)=n(1-\\beta)$. In order to make sure the output sequence of $f$, denoted by $Z$,\nis $\\epsilon$-close to $U_m$, it has\n$$\\lim_{m\\rightarrow\\infty} \\frac{m}{n(1-\\beta)}\\leq \\lim_{m\\rightarrow\\infty} \\frac{H(Z)+\\log_2\\frac{1}{1-\\epsilon}}{H(Y)}\\leq 1.$$\nSo we cannot generate more than $n(1-\\beta)$ random bits asymptotically. In this case, if we apply the seeded extractor $f$\nto the random sequence $X$, which is a possibility of $Y$, then the efficiency is\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{H(X)}=\\lim_{m\\rightarrow\\infty}\\frac{m}{n}\\leq 1-\\beta.$$\nSo there does not exist a seeded extractor that can extract randomness from an arbitrary $Y$ and its efficiency is strictly larger than $1-\\beta$.\nHere, $\\beta$ is the uncertainty of the source.\n\n\n\\begin{Theorem} \\label{var_theorem_lowerbound1} Let $\\mathcal{M}$ be a known stochastic process, and $\\mathcal{R}$ be an arbitrary stochastic process such that\n$d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then one cannot construct a variable-length extractor whose efficiency\nis strictly larger than $1-\\beta$ for all possible $\\mathcal{R}$.\n\\end{Theorem}\n\n\\proof Let $f$ be a variable-length extractor whose input sequence is a random sequence $X_m$ on $S_p$ and\nits output sequence is a random sequence $Y$ on $\\{0,1\\}^m$. Assume that as $m\\rightarrow\\infty$, $f$ can extract from an arbitrary $\\mathcal{R}$ such that the output sequence  $Y$ is $\\epsilon$-close to $U_m$.\n\nLet $h=H_{\\mathcal{M}}(X_m)$ be the entropy of the input sequence based on the distribution of $\\mathcal{M}$,\nthen we want to show that there exists a process $\\mathcal{R}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ and\n$H_{\\mathcal{R}}(X_m)\\leq h(1-\\beta)$ as $m\\rightarrow\\infty$.\n\nTo find such a process $\\mathcal{R}$, we order all the elements in $S_p$ as\n$x_1, x_2, x_3, ...$\nsuch that\n$$P_{\\mathcal{M}}(x_1)\\geq P_{\\mathcal{M}}(x_2)\\geq P_{\\mathcal{M}}(x_3)\\geq ...$$\n\nThen we divide all these elements into groups,\n$$\\{x_1, x_2, ..., x_{i_1}\\}, \\{x_{i_1+1},x_{i_1+2},...,x_{i_2}\\},...$$\nsuch that the total probability of the elements in each group is almost the probability of its first element to the power of $1-\\beta$, i.e.,\n$$0\\leq P_{\\mathcal{M}}(x_{i_j+1})^{1-\\beta}-\\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k)\\leq P_{\\mathcal{M}}(x_{i_j+1}),$$  for all  $j\\geq 0$,\nwhere $i_0=0$.\n\nLet $A=\\{x_{1}, x_{i_1+1}, x_{i_2+1}, ...\\}$ be the set consisting of the first elements of all the groups.\nNow, we consider a possibility of $\\mathcal{R}$ in the following way: for all $x\\in \\{x_{1}, x_{i_1+1}, x_{i_2+1}, ...\\}$,\nits probability is\n$$P_{\\mathcal{R}}(x)=\\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k), \\textrm{ if } x=x_{i_j+1};$$\nFor all $x\\in S_p\/A=S_p\/\\{x_{1}, x_{i_1+1}, x_{i_2+1}, ...\\}$, its probability is\n$$P_{\\mathcal{R}}(x)=0.$$\n\nFor this source $\\mathcal{R}$, the entropy of the input sequence is\n$$H_{\\mathcal{R}}(X_m)=\\sum_{x\\in S_p} P_{\\mathcal{R}}(x) \\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}.$$\n\nAs $m\\rightarrow\\infty$, we have\n\\begin{eqnarray*}\n&&H_{\\mathcal{R}}(X_m)\\\\\n   &= & \\sum_{x\\in A} P_{\\mathcal{R}}(x) \\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}  \\\\\n &\\rightarrow& (1-\\beta) \\sum_{x\\in A}  P_{\\mathcal{R}}(x) \\log_2 \\frac{1}{P_{\\mathcal{M}}(x)}\\\\\n&=& (1-\\beta)  \\sum_{j\\geq 0} \\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k) \\log_2 \\frac{1}{P_{\\mathcal{M}}(x_{i_j+1})}\\\\\n&\\leq & (1-\\beta)  \\sum_{j\\geq 0}  \\sum_{k=i_j+1}^{i_{j+1}} P_{\\mathcal{M}}(x_k) \\log_2 \\frac{1}{P_{\\mathcal{M}}(x_k)}\\\\\n&=& (1-\\beta) H_{\\mathcal{M}}(X_m)\\\\\n&=& (1-\\beta)h.\n\\end{eqnarray*}\n\nAccording to Lemma \\ref{var_lemma2}, as $m\\rightarrow\\infty$, $\\frac{m}{H_{\\mathcal{R}}(Y)}\\rightarrow 1$.\nFurthermore, we can get $$\\lim_{m\\rightarrow\\infty}\\frac{H_{\\mathcal{R}}(Y)}{H_{\\mathcal{R}}(X_m)} \\leq 1, $$\nit implies that\n$$\\lim_{m\\rightarrow\\infty} \\frac{m}{(1-\\beta)h}\\leq 1,$$\notherwise, the output sequence cannot be $\\epsilon$-close to the uniform distribution $U_m$.\n\nIf we apply the extractor $f$ to the source $\\mathcal{M}$, which is also a possibility for $\\mathcal{R}$, then its efficiency is\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{h}\\leq 1-\\beta.$$\n\nSo it is impossible to construct a variable-length extractor with efficiency strictly larger than $1-\\beta$ for all the possibilities of the source $\\mathcal{R}$. This completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nWith the same proof, we can also get the following theorem.\n\n\\begin{Theorem} Let $\\mathcal{R}$ be an arbitrary stochastic process such that\n$d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a stationary ergodic process $\\mathcal{M}$ with unknown distribution,\n, then one cannot construct a variable-length extractor whose efficiency\nis strictly larger than $1-\\beta$ for all possible $\\mathcal{R}$.\n\\end{Theorem}\n\nThe above theorems show that one cannot construct an extractor whose efficiency is strictly larger than $1-\\beta$ for all the possible source $\\mathcal{R}$. Here, $\\beta$ is an important parameter that measures the uncertainty of a real source $\\mathcal{R}$, either to\na known process or to the nearest stationary ergodic process.\nIn the next a few sections, we will present a few constructions for efficiently extracting randomness from the sources described in this section.\nWe show that their efficiency $\\eta$ satisfies\n$$1-\\beta\\leq \\eta\\leq 1.$$\nThat means the bound $1-\\beta$ is actually achievable and the constructions proposed in this paper are asymptotically optimal on efficiency.\n\n\n\\section{Construction I: Approximated by Known Processes}\n\\label{var_section_tech1}\n\nIn this section, we consider those sources which can be approximated by a known stochastic process $\\mathcal{M}$, namely, an arbitrary process\n$\\mathcal{R}$ with $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a known process $\\mathcal{M}$.\nWe say that a stochastic process $\\mathcal{M}$ is known if its distribution is given, i.e., $P_{\\mathcal{M}}(x)$ can be easily calculated for any $x\\in \\{0,1\\}^*$.\nNote that this process $\\mathcal{M}$ is not necessary to be stationary or ergodic. For instance, $\\mathcal{M}$ can be an independent process\n$z_1z_2...\\in \\{0,1\\}^*$ such that $$\\forall i\\geq1, P_{\\mathcal{M}}(z_i=1)= \\frac{1+sin(i\/10)}{2}.$$\n\n\\subsection{Construction}\n\nOur goal is to extract randomness from an imperfect random source $\\mathcal{R}$. The problem is that\nwe do not know the exact distribution of $\\mathcal{R}$, but we know that it can be approximated by a known process $\\mathcal{M}$.\nSo we can use the distribution of $\\mathcal{M}$ to estimate the distribution of $\\mathcal{R}$. As a result, we have the following procedure to extract $m$ almost-random bits.\n\nThe idea of the procedure is first producing a random sequence of length $n$ and min-entropy $k=m(1+\\alpha)$ with $\\alpha>0$, from which\nwe can further obtain a sequence $\\epsilon$-close to the uniform distribution $U_m$ by applying a $(k,\\epsilon)$  seeded extractor.\nAccording to the results of seeded extractors, this constant $\\alpha>0$ can be arbitrarily small.\n\n\\begin{Construction}\\label{const:1}\nAssume the real source $\\mathcal{R}$ is an arbitrary stochastic process such that $d(\\mathcal{R}, \\mathcal{M})\\leq \\beta$\nfor a known process $\\mathcal{M}$. Then we extract $m$ almost-random bits from $\\mathcal{R}$ based on the following procedure.\n\n\\begin{enumerate}\n  \\item Read input bits one by one from $\\mathcal{R}$ until we get an input sequence $x\\in \\{0,1\\}^*$ such that\n  $$\\log_2 \\frac{1}{P_{\\mathcal{M}}(x)}\\geq \\frac{k}{1-\\beta_p},$$\n  where  $\\beta_p=\\beta+\\epsilon_p$ with $\\epsilon_p>0$ and $k=m(1+\\alpha)$ with $\\alpha>0$. The small constant $\\epsilon_p$ has value depending on the input set $S_p$; as\n  $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. The constant $\\alpha$ can be arbitrarily small.\n  \\item  Let $n$ be the maximum length of all the possible input sequences, then\n  {$$n=\\arg\\min_{l}\\{l\\in \\mathbb{N}|\\forall y\\in \\{0,1\\}^l, $$\n  $$\\log_2\\frac{1}{P_\\mathcal{M}(y)}\\geq \\frac{k}{1-\\beta_p}\\}.$$}\n  If $|x|<n$, we  extend the length of $x$ to $n$ by adding $n-|x|$ trivial zeros at the end.\n Since $x$ is randomly generated, from the above procedure we get a random sequence $Z$ of length $n$. And it can be proved\n that this random sequence has min-entropy $k$.\n  \\item Applying a $(k,\\epsilon)$ extractor to $Z$ yields a binary sequence of length $m$ that is $\\epsilon$-close to the uniform distribution $U_m$.  \\hfill$\\Box$\n\\end{enumerate}\n\\end{Construction}\n\nThe following example is provided for comparing this construction with fixed-length constructions.\n\n\\begin{Example} Let $\\mathcal{M}$ be a biased coin with probability $0.8$ (of being $1$). If $\\frac{k}{1-\\beta_p}=2$, then we can get the input set\n$$S_p=\\{0, 10, 110, 1110, 11110,111110, 1111110, 1111111\\}.$$\nIn this case, the expected input length is strictly smaller than $7$.\nFor fixed-length constructions, to get a random sequence with min-entropy at least $2$, we have to read $7$ input bits independent of the context.\nIt is less efficient than the former method.\\hfill\n\t$\\Box$\n\\end{Example}\n\n\\begin{Theorem} Construction \\ref{const:1} generates a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\n\\end{Theorem}\n\n\\proof We only need to prove that given a source $\\mathcal{R}$ and a model $\\mathcal{M}$ with $d_p(\\mathcal{R},\\mathcal{M})\\leq \\beta_p$, it yields a random sequence $Z$ with min-entropy at least $k$.\n\nAccording to the definition of $d_p(\\mathcal{R},\\mathcal{M})$, for all $x\\in S_p$,\n$$\\frac{\\log_2\\frac{P_\\mathcal{R}(x)}{P_\\mathcal{M}(x)}}{\\log_2 \\frac{1}{P_\\mathcal{M}(x)}}\\leq \\beta_p.$$\n\nBased on the construction, for all $x\\in S_p$\n$$\\log_2\\frac{1}{P_\\mathcal{M}(x)}\\geq \\frac{k}{1-\\beta_p}.$$\n\nThe two inequalities above yield that\n$$\\log_2 \\frac{1}{P_\\mathcal{R}(x)}\\geq k,$$\nfor all $x\\in S_p$.\n\nSince the second step, i.e., adding trivial zeros, does not reduce the min-entropy of $S_p$. As\na result, we get a random sequence $Z$ of length $n$ and with min-entropy at least $k$.\n\nSince $k=m(1+\\alpha)$ with $\\alpha>0$, according to Lemma \\ref{lemma_seededextractor},\nwe can construct a seeded extractor that applies to the sequence $Z$ and generates a binary sequence $\\epsilon$-close to the uniform distribution $U_m$.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\n\\subsection{Efficiency Analysis}\n\nNow, we study the efficiency of Construction \\ref{const:1}. According to our definition, given\na construction, its efficiency is\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{H_{\\mathcal{R}}(X_m)}.$$\n\n\\begin{Theorem}\\label{var_theorem1_2} Given a real source $\\mathcal{R}$ and a known process $\\mathcal{M}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then the efficiency of Construction \\ref{const:1} is\n$$1-\\beta\\leq \\eta\\leq 1.$$\n\\end{Theorem}\n\n\\proof Since $\\eta$ is always upper bounded by $1$, we only need to show that $\\eta\\geq 1-\\beta$.\n\n According to Lemma \\ref{lemma_seededextractor}, as $m\\rightarrow \\infty$,\nwe have\n$$\\lim_{m\\rightarrow\\infty}\\frac{k}{m}=1.$$\n\nNow, let us consider the number of elements in $S_p$, i.e., $|S_p|$. To calculate $|S_p|$, we let\n$$S_p'=\\{x[1:|x|-1] |x\\in S_p\\},$$\nwhere $x[1:|x|-1]$ is the prefix of $x$ of length $|x|-1$,\nthen for all $y\\in S_p'$, $$\\log_2\\frac{1}{P_\\mathcal{M}(y)}\\leq \\frac{k}{1-\\beta_p}.$$  Hence,\n$$\\log_2|S_p'|\\leq \\frac{k}{1-\\beta_p}.$$\n\nIt is easy to see that $|S_p|\\leq 2|S_p'|$, so\n$$\\log_2|S_p|\\leq \\frac{k}{1-\\beta_p}+1.$$\n\nLet $X_m$ be the input sequence, then\n$$\\lim_{k\\rightarrow\\infty}\\frac{H_{\\mathcal{R}}(X_m)}{k}\\leq \\lim_{k\\rightarrow\\infty}\\frac{\\log_2|S_p|}{k}$$\n$$\\leq \\lim_{k\\rightarrow\\infty}\\frac{1}{1-\\beta_p}=\\frac{1}{1-\\beta}.$$\n\nFinally, it yields\n$$\\eta=\\lim_{m\\rightarrow\\infty}\\frac{m}{H_{\\mathcal{R}}(X_m)}\\geq 1-\\beta.$$\n\nThis completes the proof.\\hfill\\IEEEQED\\vspace{0.1in}\n\nWe see that the efficiency of the above construction is between $1-\\beta$ and $1$. As shown in Theorem \\ref{var_theorem_lowerbound1}, the\ngap $\\beta$, introduced by the uncertainty of the real source $\\mathcal{R}$, cannot be smaller.  Our construction is asymptotically optimal in the sense that we cannot find a variable-length extractor with efficiency definitely larger than $1-\\beta$.\n\n\\begin{Corollary} Given a real source $\\mathcal{R}$ and a known process $\\mathcal{M}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then as $\\beta\\rightarrow 0$, the efficiency of Construction \\ref{const:1} is\n$$\\eta\\rightarrow 1.$$\n\\end{Corollary}\n\nIn this case, the efficiency of the construction can achieve Shannon's limit.\n\nIf $\\mathcal{R}$ is a stationary ergodic process, we can also get the following result.\n\n\\begin{Corollary}  Given a stationary ergodic process$\\mathcal{R}$ and a known process $\\mathcal{M}$ such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, for the expected input length of Construction \\ref{const:1}, we have\n$$\\frac{1}{h(\\mathcal{R})}\\leq \\lim_{m\\rightarrow\\infty} \\frac{E[|X_m|]}{m}\\leq \\frac{1}{(1-\\beta)h(\\mathcal{R})},$$\nwhere $h(\\mathcal{R})$ is the entropy rate of the source $\\mathcal{R}$.\n\\end{Corollary}\n\n\\proof This conclusion is immediate following Lemma \\ref{var_theorem_stionaryergodiclength}  and Theorem \\ref{var_theorem1_2}.\n\\hfill\n\\IEEEQED\\vspace{0.1in}\n\n\\section{Construction II: Approximately Biased Coins}\n\\label{var_section_tech2}\n\nIn this section, we use a general ideal model such as a biased coin or a Markov chain to approximate the real source $\\mathcal{R}$.\nHere, we do not care about the specific parameters of the ideal model. The reason is, in some cases, the source $\\mathcal{R}$ is\nvery close to an ideal source but we cannot (or do not want to) estimate the parameters accurately.\nAs a result, we introduce a construction by exploring the characters of biased coins or Markov chains. For simplicity,\nwe only discuss the case that\nthe ideal model is a biased coin, and the same idea can be generalized when the ideal model is a Markov chain. Specifically,\nlet $\\mathcal{G}_{b.c.}$ denote the set consisting of all the models of biased coins with different probabilities, and we\nconsider $\\mathcal{R}$ as an arbitrary stochastic process such that\n$$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta.$$\n\n\\subsection{Construction}\n\nThe idea of the construction is similar as Construction \\ref{const:1}, i.e., we  first produce a random sequence of length $n$ and with min-entropy $k=m(1+\\alpha)$ for $\\alpha>0$, from which\nwe can further obtain a sequence $\\epsilon$-close to the uniform distribution $U_m$ by applying a $(k,\\epsilon)$  seeded extractor.\n\n\\begin{Construction}\\label{const:2}\nAssume the real source $\\mathcal{R}$ is an arbitrary stochastic process such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta$$ for a constant $\\beta$.   Then we extract $m$ almost-random bits from $\\mathcal{R}$ based on the following procedure.\n\\begin{enumerate}\n\\item Read input bits one by one from $\\mathcal{R}$ until we get an input sequence $x\\in \\{0,1\\}^*$ such that\n     $$\\log_2 {\\nchoosek{k_0+k_1}{ \\max(1,\\min(k_0,k_1))}}\\geq \\frac{k}{1-\\beta_p},$$\n     where $k_0$ is the number of zeros in $x$, $k_1$ is the number of ones in $x$,\n    $\\beta_p=\\beta+\\epsilon_p$ with $\\epsilon_p>0$ and $k=m(1+\\alpha)$ with $\\alpha>0$. The small constant $\\epsilon_p$ has value depending on the input set $S_p$; as\n  $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. The constant $\\alpha$ can be arbitrarily small.\n\\item Since the input sequence $x$ can be very long, we map it into a sequence $z$ of fixed length $n$ such that\n$$z=[I_{(k_0\\geq k_1)}, \\min(k_0,k_1), r(x)],$$\nwhere $I_{(k_0\\geq k_1)}=1$ if and only if $k_0\\geq k_1$, and $r(x)$ is the rank of $x$  among all the permutations of $x$ with respect to the lexicographic order.\nSince $x$ is randomly generated, the above procedure leads us to a random sequence $Z$ of length $n$.\n\\item Applying a $(k,\\epsilon)$ extractor to $Z$ yields a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\\hfill$\\Box$\n\\end{enumerate}\n\\end{Construction}\n\nTo see that the construction above works, we need to show that the random sequence $Z$ obtained after the second step has min-entropy at least $k$, and its length $n$ is well bounded.\n\n\\begin{Lemma}\n Given a source $\\mathcal{R}$ with $\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta$, Construction \\ref{const:2} yields a random sequence $Z$ with length\n $$n\\leq 1+\\lceil\\log_2(\\frac{k}{1-\\beta_p}+1)\\rceil+ \\lceil\\frac{2k}{1-\\beta_p}\\rceil.$$\n\\end{Lemma}\n\\proof\n1) $I_{(k_0\\geq k_1)}$ can be represented as $1$ bit.\n\n2) Without loss of generality, we assume $k_0\\leq k_1$. According to our construction,\n$$ \\log_2 {\\nchoosek{k_0+k_1-1}{ k_0-1}}< \\frac{k}{1-\\beta_p} \\textrm{ for } k_0>1,$$\nand\n$$ \\log_2 {\\nchoosek{k_1}{1}} < \\frac{k}{1-\\beta_p} \\textrm{ for } k_0=0 \\textrm{ or } k_0=1.$$\n\nThen\n\\begin{eqnarray*}\nk_0-1&\\leq &\\log_2 {\\nchoosek{2k_0-1}{ k_0-1}}\\\\\n&\\leq& \\log_2 {\\nchoosek{k_0+k_1-1}{k_0-1}}\\\\\n&<& \\frac{k}{1-\\beta_p}.\n\\end{eqnarray*}\n\nSo $\\min(k_0, k_1)$ can be represented as\n$\\lceil\\log_2 (\\frac{k}{1-\\beta_p} +1 )\\rceil$ bits.\n\n3) Let us consider the number of permutations of $x$, denoted by $N(x)$.\nIf $k_0>1$, then\n\\begin{eqnarray*}\n\\log_2 N(x)&=&\\log_2 {\\nchoosek{k_0+k_1}{ k_0}}\\\\\n&\\leq &\\log_2  {\\nchoosek{k_0+k_1-1}{ k_0-1}} +\\log_2\\frac{k_0+k_1}{k_0}\\\\\n&\\leq &\\frac{k}{1-\\beta_p} +\\log_2\\frac{k_0+k_1}{k_0}.\n\\end{eqnarray*}\n\nIf $k_0=1$, then\n$$\\log_2 N(x) \\leq \\log_2  {\\nchoosek{k_1 }{1}} +\\log_2 \\frac{k_1+1}{k_1}.$$\n\nIf $k_0=0$, then\n$$\\log_2 N(x)=0.$$\n\nBased on the analysis above, we can get $$\\log_2 N(x)\\leq \\frac{2k}{1-\\beta_p}.$$\n\nHence, $r(x)$ can be represented as $\\lceil\\frac{2k}{1-\\beta_p}\\rceil$ bits.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nLet $\\mathbf{1}^a$ denote the all-one vector of length $a$, then we get the following result.\n\\begin{Theorem}\n Construction \\ref{const:2} generates a random sequence of length $m$ that is $\\epsilon$-close to $U_m$ if\n $P_{\\mathcal{R}}(\\mathbf{1}^a)\\leq 2^{-k}, P_{\\mathcal{R}}(\\mathbf{0}^a)\\leq 2^{-k}$ for $a=2^{\\lfloor\\frac{k}{1-\\beta_p}\\rfloor}$.\n\\end{Theorem}\n\n\\proof Since the mapping in the second step is injective, it will not affect the min-entropy; we only need to prove that the input sequence has min-entropy $k$, i.e.,\n$$\\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}\\geq k, \\forall x\\in S_p,$$\nwhere $S_p$ is the set consisting of all the possible input sequences.\n\nIt is not hard to see that if $\\min(k_0,k_1)\\geq 1$,\n$$P_{\\mathcal{M}}(x)\\leq \\frac{1}{{\\nchoosek{k_0+k_1}{ k_0}}},$$\nwhich leads to\n$$\\log_2 \\frac{1}{P_{\\mathcal{M}}(x)}\\geq \\frac{k}{1-\\beta_p}.$$\n\nFurthermore, based on the definition of $d_p(\\mathcal{R},\\mathcal{M})$, we can get if $\\min(k_0,k_1)\\geq 1$,\n$$\\log_2 \\frac{1}{P_{\\mathcal{R}}(x)}\\geq k.$$\n\nIf $\\min(k_0,k_1)=0$, according to the condition in the lemma, we can also have the same result.\n\nSince $k=m(1+\\alpha)$ with $\\alpha>0$, according to Lemma \\ref{lemma_seededextractor},\nwe can construct a seeded extractor that applies to the sequence $Z$ and generates a binary sequence $\\epsilon$-close to the uniform distribution $U_m$.\n\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nActually, the idea above can be easily generalized if $\\mathcal{M}$ is a Markov chain that best approximates the real source $\\mathcal{R}$.\nThe idea follows the main lemma in \\cite{Zhou_Markov} that shows how to generate random bits with optimal efficiency from an arbitrary Markov chain.\n\n\\subsection{Efficiency Analysis}\n\nFor the efficiency of the construction, we can get the same bounds as Construction \\ref{const:1}.\n\n\\begin{Theorem}\nGiven an arbitrary source $\\mathcal{R}$ such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}}d (\\mathcal{R},\\mathcal{M}) \\leq \\beta,$$\nif there exists a model $\\mathcal{M}\\in \\mathcal{G}_{b.c.}$ with probability $p\\leq \\frac{1}{2}$ of being $1$ or $0$ and\n$$p>\\sqrt{d(\\mathcal{R},\\mathcal{M})\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}},$$ then the efficiency of Construction \\ref{const:2} is\n$$1-\\beta \\leq \\eta\\leq 1.$$\n\\end{Theorem}\n\n\\proof Let $N_{k_0, k_1}$ denote the number of input sequences with $k_0$ zeros and $k_1$ ones in $S_p$, and let $p_{k_0,k_1}$ be the probability based on $\\mathcal{R}$ of generating such a sequence. Let us define\n$$A=\\{(k_0,k_1)| N_{k_0,k_1}>0\\},$$\nthen we can get\n$$\nH_{\\mathcal{R}}(X_m)\\leq H(\\{p_{k_0,k_1}|(k_0,k_1)\\in A\\})$$\n$$+\\sum_{(k_0,k_1)\\in A}p_{k_0,k_1}\\log_2 N_{k_0,k_1}.\n$$\n\nAccording to the proof in the above theorem, $\\min(k_0,k_1)\\leq \\frac{k}{1-\\beta_p}+1$. So there are totally at most $2(\\frac{k}{1-\\beta_p}+1)$ available pairs of $(k_0, k_1)$. Hence\n$$H(\\{p_{k_0,k_1}|(k_0,k_1)\\in A\\})\\leq \\log_2(2+(\\frac{k}{1-\\beta_p}+1))=o(k).$$\n\nNow, we write $n=k_0+k_1$. According to our method, if $\\min(k_0,k_1)\\geq 1$,\n$$\\nchoosek{k_0+k_1}{\\min(k_0,k_1)}\\geq 2^{\\frac{k}{1-\\beta_p}},$$\n$$\\nchoosek{k_0+k_1-1}{\\min(k_0,k_1)-1}<2^{\\frac{k}{1-\\beta_p}}.$$\n\nHence, given $n$, we get an upper bound for $\\min(k_0,k_1)$, which is\n\\begin{equation}\\label{var_equ_lemma1_1}\nt_n=\\max\\{i\\in\\{0,1,...,n\\}| {\\nchoosek{n-1}{ i-1}}<2^{\\frac{k}{1-\\beta_p}}\\}.\n\\end{equation}\n\nNote that if $\\nchoosek{n-1}{\\frac{n}{2}-1}\\geq 2^{\\frac{k}{1-\\beta_p}}$, then $t_n$ is a nondecreasing\nfunction of $n$. Using the Stirling bounds on factorials yields\n$$ \\lim_{n\\rightarrow \\infty }\\frac{1}{n}\\log_2 {\\nchoosek{n}{ \\rho n}}=H(\\rho),$$\nwhere $H$ is the binary entropy function. Hence, following (\\ref{var_equ_lemma1_1}), we can get\n\\begin{equation}\\label{var_equ_lemma1_3} \\lim_{n\\rightarrow \\infty} H(\\frac{t_n}{n})=\\lim_{n\\rightarrow\\infty}\\frac{k}{(1-\\beta_p) n}.\\end{equation}\n\nLet $P_n$ denote the probability of having an input sequence of length at least $n$ based on the distribution of $\\mathcal{R}$. In this case,\n$P_n$ is a nonincreasing function of $n$. Let $Q_n$ denote the probability of having an input sequence of length at least $n$ based on the distribution of $\\mathcal{M}\\in \\mathcal{G}_{b.c.}$\nwhose probability is $p\\leq \\frac{1}{2}$. Since for all binary sequence $x\\in \\{0,1\\}^n$,\n$$\\log_2 \\frac{1}{P_\\mathcal{M}(x)}\\leq n\\log_2\\frac{1}{p},$$\nwe can get\n$$\\log_2\\frac{P_{\\mathcal{R}(x)}}{P_\\mathcal{M}(x)}\\leq d n\\log_2 \\frac{1}{p},$$\nwhere $d=d_p(\\mathcal{R},\\mathcal{M})$.\n\nSince\n$P_n=\\sum_{x\\in S} P_{\\mathcal{R}}(x)$ and $Q_n=\\sum_{x\\in S}\nP_{\\mathcal{M}}(x)$ for some $S\\subset \\{0,1\\}^n$, it is not hard to prove that\n\\begin{equation}\\label{equ_lemma1_13}\\log_2\\frac{P_n}{Q_n}\\leq d n \\log_2\\frac{1}{p}.\\end{equation}\n\nAccording to Hoeffding's inequality, we can get\n\\begin{eqnarray*}\nQ_n&\\leq &2P[k_1\\leq t_n]\\\\\n&\\leq& 2P[\\frac{k_1}{n}-p \\leq \\frac{t_n}{n}-p] \\\\\n&\\leq& 2e^{-2n(p-\\frac{t_n}{n})^2}.\n\\end{eqnarray*}\n\nHence\n\\begin{equation}\\label{equ_lemma1_14}P_n \\leq 2^{- d n\\log_2 p} Q_n \\leq 2 e^{-\\log_2 p \\ln 2 \\cdot d n  -2 n (p-\\frac{t_n}{n})^2}.\\end{equation}\n\nFrom this inequality, we see that $P_n\\rightarrow 0$ as $n\\rightarrow 0$ if\n\\begin{equation}\\label{equ_lemma1_15}-d \\log_2 p \\ln 2 -2 (p-\\frac{t_n}{n})^2 <0.\\end{equation}\n\nBased on (\\ref{var_equ_lemma1_3}) and (\\ref{equ_lemma1_15}), we can get that $P_n\\rightarrow 0$ as $n\\rightarrow 0$ if\n$$\\frac{n}{k}\\geq \\frac{1}{(1-\\beta_p)H(p-\\sqrt{d\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}})}.$$\n\nNow, let $a=\\frac{1+\\epsilon}{(1-\\beta_p)H(p-\\sqrt{d\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}})}$ with $\\epsilon>0$, we can write\n\\begin{eqnarray*}\nH_{\\mathcal{R}}(X_m) &\\leq &o(k)+ \\sum_{k_0,k_1: k_0+k_1\\geq a k} p_{k_0,k_1}\\log_2 N_{k_0,k_1}\\\\\n&&+ \\sum_{k_0,k_1: k_0+k_1<a k} p_{k_0,k_1}\\log_2 N_{k_0,k_1}.\n\\end{eqnarray*}\n\nAccording to our analysis, if $k_0+k_1\\geq a k$, as $k\\rightarrow \\infty$, $$P_n=\\sum_{k_0,k_1:k_0+k_1\\geq a k} p_{k_0,k_1}\\rightarrow 0$$ and\n$\\log_2 N_{k_0,k_1}\\leq 2\\frac{k}{1-\\beta_p}$.  If $k_0+k_1\\leq a k$, then $$\\log_2 N_{k_0,k_1}\\leq \\frac{k}{1-\\beta_p}+\\log_2 \\frac{k_0+k_1}{\\min (k_0,k_1)}\\leq \\frac{k}{1-\\beta_p} +o(k).$$ As a result, we can get\n\\begin{eqnarray*}\nH_{\\mathcal{R}}(X_m)& \\leq &o(k)+ o(1)\\frac{2k}{1-\\beta_p} + (\\frac{k}{1-\\beta_p}+o(k))\\\\\n&\\leq& \\frac{k}{1-\\beta_p}+o(k).\n\\end{eqnarray*}\n\nSo\n$$\\lim_{k\\rightarrow\n\\infty}\\frac{k}{H_{\\mathcal{R}}(X_m)}\\geq 1-\\beta.$$\n\nFurthermore, based on the fact that $\\lim_{m\\rightarrow\\infty}\\frac{k}{m}=1$, we can get $\\eta\\geq 1-\\beta$.\nIt is known that $\\eta\\leq 1$, so it concludes the theorem.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nSimilar to Construction \\ref{const:1}, this construction is also asymptotically optimal in the sense\nthat we cannot find a variable-length extractor with efficiency definitely larger than $1-\\beta$, as shown in Theorem \\ref{var_theorem_lowerbound1}.\n\n\\begin{Corollary} Given an arbitrary source $\\mathcal{R}$ such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta,$$ then as $\\beta\\rightarrow 0$, the efficiency of Construction \\ref{const:2} is\n$$\\eta\\rightarrow 1.$$\n\\end{Corollary}\n\nIt is easy to see that as $\\beta\\rightarrow 0$, Construction \\ref{const:2} reaches the Shannon's limit on efficiency. If\n$\\mathcal{R}$ is a stationary ergodic process, we can also get the following corollary.\n\n\\begin{Corollary}\nGiven an arbitrary stationary ergodic source $\\mathcal{R}$ such that $$\\min_{\\mathcal{M}\\in \\mathcal{G}_{b.c.}} d(\\mathcal{R},\\mathcal{M}) \\leq \\beta,$$\nif there exists a model $\\mathcal{M}\\in \\mathcal{G}_{b.c.}$ with probability $p\\leq \\frac{1}{2}$ of being $1$ or $0$ and\n$$p>\\sqrt{d(\\mathcal{R},\\mathcal{M})\\log_2\\frac{1}{p}\\frac{\\ln 2}{2}},$$ then for the expected input length of Construction \\ref{const:2}, we have\n$$\\frac{1}{h(\\mathcal{R})}\\leq \\lim_{m\\rightarrow\\infty} \\frac{E[|X_m|]}{m}\\leq \\frac{1}{(1-\\beta)h(\\mathcal{R})},$$\nwhere $h(\\mathcal{R})$ is the entropy rate of $\\mathcal{R}$.\n\\end{Corollary}\n\n\n\n\n\n\\section{Construction III: Approximately Stationary Ergodic Processes}\n\\label{var_section_tech4}\n\n\nIn this section, we consider imperfect sources that are approximately stationary and ergodic.\nHere, we let  $\\mathcal{R}$ be an arbitrary stochastic process such that $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$ for a stationary ergodic process $\\mathcal{M}$.\nFor these sources, universal data compression can be used to `purify' input sequences, i.e., shortening their lengths while maintaining their entropies.\nIn \\cite{Visweswariah98}, Visweswariah, Kulkarni and Verd\\'{u} showed that optimal variable-length source codes asymptotically achieve optimal variable-length random bits generation in the sense of normalized divergence. Although\ntheir work only focused on ideal stationary ergodic processes and generates `weaker' random bits, it motivates us to combine\nuniversal compression with fixed-length extractors for efficiently generating random bits from noisy stochastic processes.\nIn this section, we will first introduce Lempel-Ziv code and then present its application in constructing variable-length extractors.\n\n\\subsection{Construction}\n\nLempel-Ziv code is a universal data compression scheme introduced by Ziv and Lempel \\cite{Ziv78}, which is simple to implement and can achieve the asymptotically optimal rate\nfor stationary ergodic sources. The idea of Lempel-Ziv code is to parse the source sequence into strings that have not appeared so far, as demonstrated by the following example.\n\n\\begin{Example} Assume the input is $010111001110000...$, then we parse it as strings\n$$0,1,01,11,00,111,000,...$$\nwhere each string is the shortest string that never appear before. That means all its prefixes have occurred earlier.\n\nLet $c(n)$ be the number of strings obtained by parsing a sequence of length $n$. For each string, we\ndescribe its location with $\\log c(n)$ bits. Given a string of length $l$, it can described by (1)\nthe location of its prefix of length $l-1$, and (2) its last bit. Hence, the code for the above sequence is\n{ $$(000,0), (000,1), (001,1), (010,1), (001,0), (100,1), (101,0),...$$}\nwhere the first number in each pair indicates the prefix location and the second number is\nthe last bit of the string.\n\\end{Example}\n\\vspace{-0.25cm}\n\\hfill$\\Box$\n\nTypically, Lempel-Ziv is applied to an input sequence of fixed length. Here, we are interested in\nLempel-Ziv code with fixed output length and variable input length. As a result, we can apply\na single fixed-length extractor to the output of Lempel-Ziv code for extracting randomness.\nIn our algorithm, we read raw bits one by one from an imperfect source until the length of the output\nof a Lempel-Ziv code reaches a certain length.\nIn another word,\nthe number of strings after parsing is a predetermined number $c$. For example, if\nthe source is $1011010100010...$ and $c=4$, then after reading $6$ bits, we can parse them into\n$1, 0, 11, 01$. Now, we get an output sequence\n$(000,1), (000,0), (001, 1), (010, 1)$, which can be used as the input of a fixed-length extractor.\nWe call this Lempel-Ziv code as a variable-length Lempel-Ziv code.\n\nLet $Z$ be a random sequence obtained based on variable-length Lempel-Ziv code such that its length is\n$$|Z|=(\\log c+1) c,$$\nfor a predetermined $c$. Then $Z$ is very close to truly random bits in the term of min-entropy if the source $\\mathcal{R}$ is stationary ergodic.\nAs a result, we have the following construction for variable-length extractors.\n\n\\begin{Construction}\\label{const:3}\nAssume the real source is $\\mathcal{R}$ and there exists a stationary ergodic process $\\mathcal{M}$ such that $d(\\mathcal{R},M)\\leq \\beta$.\nThen we extract $m$ almost random bits from $\\mathcal{R}$ based on the following procedure.\n\\begin{enumerate}\n\\item Read input bits one by one based on the variable-length Lempel-Ziv code until we get an output sequence $Z$ whose length reaches $$n=\\frac{k}{1-\\beta_p}(1+\\varepsilon),$$\n    where $\\varepsilon>0$ is a small constant indicating the performance gap between the case of finite-length and that of infinite-length\n    for Lempel-Ziv code; as $m\\rightarrow\\infty$, we have $\\varepsilon\\rightarrow 0$. Similar as above,\n    $\\beta_p=\\beta+\\epsilon_p$ with $\\epsilon_p>0$ and $k=m(1+\\alpha)$ with $\\alpha>0$. The small constant $\\epsilon_p$ has value depending on the input set $S_p$; as\n  $m\\rightarrow\\infty$, $\\epsilon_p\\rightarrow 0$. The constant $\\alpha$ can be arbitrarily small.\n    Then we get a random sequence $Z$ of length $n$ and with min-entropy $k$.\n\\item Applying a $(k,\\epsilon)$ extractor to $Z$ yields a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\\hfill$\\Box$\n\\end{enumerate}\n\\end{Construction}\n\nWe show that the min-entropy of $Z$ is at least $k$ as $m\\rightarrow\\infty$.  If $m$ is not very large, by adjusting the parameter $\\varepsilon$,\nwe can make the min-entropy of $Z$ be at least $k$.\nSo we can continue to apply\nan efficient fixed-length extractor to `purify' the resulting sequence. Finally, we can get $m$\nrandom bits that satisfy our requirements on quality in the sense of statistical distance.\n\n\\begin{Theorem}\\label{theorem_3_1}\nWhen $m\\rightarrow\\infty$, Construction \\ref{const:3} generates a random sequence of length $m$ that is $\\epsilon$-close to $U_m$.\n\\end{Theorem}\n\n\\proof Let $x$ be an input sequence. According to theorem 12.10.1 in \\cite{Cover2006}, for the stationary ergodic process $\\mathcal{M}$,\nwe can get\n$$\\frac{1}{|x|}\\log_2 \\frac{1}{ P_{\\mathcal{M}}(x)}\\geq \\frac{c}{|x|}\\log_2 c -\\frac{c}{|x|} H(U,V),$$\nwhere  $$\\frac{c}{|x|}H(U,V)\\rightarrow 0 \\textrm{ as } |x|\\rightarrow 0.$$\n\nAs a result, if $k=\\Theta(n)$,\n\\begin{eqnarray*}\n\\lim_{k\\rightarrow\\infty}\\frac{1}{k}\\log_2 \\frac{1}{ P_{\\mathcal{R}}(x)}&\\geq& \\lim_{k\\rightarrow\\infty}(1-\\beta_p)\\frac{1}{k}\\log_2\\frac{1}{P_\\mathcal{M}(x)}\\\\\n& \\geq &\\lim_{k\\rightarrow\\infty} \\frac{(1-\\beta_p)c\\log_2 c}{k} \\\\\n&=&  \\lim_{k\\rightarrow \\infty}\\frac{(1-\\beta_p)n}{k}\\\\\n&=& \\lim_{k\\rightarrow\\infty} 1+\\varepsilon\\\\\n&=& 1.\n\\end{eqnarray*}\n\nFinally, we can get that\n$$\\lim_{k\\rightarrow\\infty} \\frac{H_{\\min}(Z)}{k}=\\lim_{k\\rightarrow\\infty} \\frac{H_{\\min}(X_m)}{k}\\geq 1.$$\nThis implies that as $m\\rightarrow \\infty$, i.e., $k\\rightarrow \\infty$, the min-entropy of $Z$ is at least $k$.\n\n\nSince $k=m(1+\\alpha)$ for an $\\alpha>0$,  we can continue to apply a $(k,\\epsilon)$ extractor to extract $m$ almost-random bits from $Z$.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\n\\subsection{Efficiency Analysis}\n\nNow, we study the efficiency of the construction based on variable-length Lempel-Ziv codes.\n\n\\begin{Theorem}\nGiven a real source $\\mathcal{R}$ such that there exists a stationary ergodic process $\\mathcal{M}$ with $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then the efficiency of Construction \\ref{const:3} is \\vspace{-0.05cm}\n$$1-\\beta\\leq \\eta\\leq 1.$$\n\\end{Theorem}\n\n\\proof Similar as above, we only need to prove that $\\eta\\geq 1-\\beta$.\n\nSince there are at most $n=2^{c(\\log_2c+1)}$ distinct input sequences,\ntheir entropy\n$$H_{\\mathcal{R}}(X_m)\\leq c(\\log_2 c+1)=n.$$\n\nAccording to the proof in Theorem \\ref{theorem_3_1}, we have that the random sequence $Z$ has min-entropy at least $k$, and it satisfies\n$$\\lim_{m\\rightarrow\\infty}\\frac{n}{k}=\\frac{1}{1-\\beta}.$$\n\nBased on the construction of seeded extractors, we can also get\n$$\\lim_{m\\rightarrow\\infty}\\frac{m}{k}=1.$$\n\nAs a result,\n$$ \\eta= \\lim_{m\\rightarrow\\infty}\\frac{m}{H_{\\mathcal{R}}(X_m)} \\geq 1-\\beta.$$\nThis completes the proof.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nAlthough Construction \\ref{const:3} has the same efficiency as the other constructions, when $m$ is not large,\nit is  less efficient than the other constructions because the Lempel-Ziv code does not always have the best performance when the input sequence is not long.\nBut its advantage is that it can manage more general sources without accurate estimations. In the above theorem, the gap $\\beta$ represents how far the source $\\mathcal{R}$ is from being stationary ergodic. In general, the efficiency loss introduced by the uncertainty of sources\nis a part that cannot be avoid.\n\n\\begin{Corollary} Given a real source $\\mathcal{R}$ such that there exists a stationary ergodic model $\\mathcal{M}$ with $d(\\mathcal{R},\\mathcal{M})\\leq \\beta$, then as $\\beta\\rightarrow 0$, the efficiency of Construction \\ref{const:3} is\n$$\\eta\\rightarrow 1.$$\n\\end{Corollary}\n\nIt shows that as $\\beta\\rightarrow 0$, Construction \\ref{const:3} reaches the Shannon's limit on efficiency.\n\n\\begin{Corollary} Given a stationary ergodic source $\\mathcal{R}$ (assume we do not know that it is stationary ergodic), for the expected input length of Construction \\ref{const:3}, we have\n$$\\frac{1}{h(\\mathcal{R})}\\leq \\lim_{m\\rightarrow\\infty} \\frac{E[|X_m|]}{m}\\leq \\frac{1}{(1-\\beta)h(\\mathcal{R})},$$\nwhere $h(\\mathcal{R})$ is the entropy rate of $\\mathcal{R}$.\n\\end{Corollary}\n\n\n\\section{Seedless Constructions}\n\\label{sec_randomnessextraction}\n\nTo simulate seeded constructions of variable-length extractors in randomized applications, we have to enumerate all possible assignments of the seed, hence,\nthe computational complexity will be increased significantly. In real applications, we prefer seedless constructions rather than seeded constructions. It motivates us to study the seedless constructions of variable-length extractors in this section.\n\n\\subsection{An Independent Source}\n\nLet us first consider a simple independent source described in the introduction.  This type of sources have been widely studied in seedless constructions of fixed-length extractors.\n\n\\begin{Example} Let $x_1x_2...\\in\\{0,1\\}^*$ be an independent sequence generated from a source $\\mathcal{R}$ such that\n$$P[x_i=1]\\in [0.9,0.91] \\quad \\forall i\\geq i.$$\n\\end{Example}\n\\vspace{-0.4cm} \\hfill$\\Box$\n\\vspace{0.2cm}\n\nWe see that the existing methods for generating random bits from ideal sources (like biased coins or Markov chains) cannot be applied here, since\nthe probability of each bit is slightly unpredictable. Some seedless extractors have been developed for extracting randomness from such sources.\nIn particular, there exists seedless extractors which are able to extract as many as $H_{\\min}(X)$ random bits from a independent random sequence $X$ asymptotically. In order to extract $m$ random bits in the above example, it needs to read $\\frac{m}{\\log_2\\frac{1}{0.91}}$ input bits as\n$m\\rightarrow\\infty$. In this case, the entropy of the input sequence is in $$[H(0.9)\\frac{m}{\\log_2\\frac{1}{0.91}}, H(0.91)\\frac{m}{\\log_2\\frac{1}{0.91}}].$$ From which, we can get the efficiency of an optimal fixed-length  extractor, which is\n$$\\eta_{fixed}\\in [0.2901, 0.3117],$$\ni.e., about only $0.3$ of the input entropy is used for generating random bits, which is far from optimal\n\nIn the above example, we let $\\mathcal{M}$ be a biased coin model with probability $p=0.9072$. In this case,\n$$\\beta\\leq d(\\mathcal{R},\\mathcal{M})=0.0315.$$\nAccording to the constructions in the previous sections, there exists seeded variable-length extractors such that\ntheir efficiencies are\n$$\\eta_{variable}\\in [1-\\beta, 1]\\subseteq [0.9685,1],$$\nwhich are near Shannon's limit.\n\nBased on the fact that the source is independent, we can eliminate the requirement of truly random bits as the seed, hence, we have seedless\nvariable-length extractors.\nTo construct a seedless variable-length extractor, we first apply a seedless fixed-length extractor $E_1$ (which may not be very efficient)\nto extract a random sequence of length $d$ from input bits.\nUsing this random sequence as the seed, we continue to apply a seeded variable-length extractors $E_2$ to\nextract $m$ almost-random bits from extra input bits. So seedless variable-length extractors can be constructed as cascades of seedless fixed-length extractors and seeded variable-length extractors. Since the input length of $E_1$ is much shorter (it is ignorable) than the input length\nof $E_2$, the efficiency of the resulting seedless extractor, i.e., $E=E_2\\bigotimes E_1$, is dominated by the efficiency of $E_2$.\nSo the efficiency of the seedless extractor $E$ is in $[0.9685,1]$,\nwhich is very close to the optimality.\n\nThis example demonstrates a simple construction of seedless variable-length extractors for independent sources, and it shows\nthe significant performance gain of variable-length extractors compared to fixed-length extractors.\n\n\\subsection{Generalized Sources}\n\n\nHere we consider a generalization of independent processes. Given a system, we use $\\lambda_i$ denote the complete system status at time $i$.\nFor example, in a system that generates thermal noise, the system status can include the value of the noise signal, the temperature, the environmental effects, etc. Usually, the evolution of such a system has a Markov property, namely,\n$$P[\\lambda_{i+1},\\lambda_{i+2},...|\\lambda_i, \\lambda_{i-1},..., \\lambda_1]=P[\\lambda_{i+1},\\lambda_{i+2},...|\\lambda_i],$$\nfor all $i\\geq 1$. Let $X=x_1x_2...\\in \\{0,1\\}^n$ be the binary sequence generated from this system, then for any $1< k<n-1$,\n\\begin{equation}\\label{equ_seedless_1}P[X_1^{k-1},X_{k+1}^{n}|\\lambda_k]=P[X_1^{k-1}|\\lambda_k]P[X_{k+1}^{n}|\\lambda_k],\\end{equation}\nwhere $X_a^b=x_ax_{a+1}...x_{b}$. In some sense, the source $X$ that we consider is a hidden Markov process, but the number of hidden states can be infinite ($\\lambda_i$ can be discrete or continuous).\n\n\\begin{Example} One example of the above sources is the one studied in \\cite{Kamp11}, called a space $s$ source.\nA space $s$ source is basically a source generated by a width $2^s$ branching program. At each step, the state of the process\ngenerating the source is in one of $2^s$ states, and the bit generated is a function of the current state. Unlike perfect Markov chains,\nthe transition probabilities can be different at each step. In this example, the system status $\\lambda_i$ is the content of space $s$ at time $i$, that is, one of the $2^s$ states, and $x_i\\in \\{0,1\\}$ is a function of $\\lambda_i$.\n\\end{Example}\n\nSpace $s$ sources are very general in that most other classes of sources that have been considered\npreviously can be computed with a small amount of space \\cite{Kamp11}. The model that we consider, as described by (\\ref{equ_seedless_1}),\nis a natural generalization of space $s$ sources. This model has a very nice feature: from such a source, we can get a group of sequences conditionally independent of each other. Namely,\ngiven system statues at some time points\n$$[\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}]=[\\lambda_{a}, \\lambda_{2a}, ..., \\lambda_{\\gamma a}],$$\nthe subsequences\n$$[X^{(1)},X^{(2)},...,X^{(\\gamma)}, X^{(\\gamma+1)}]$$\n$$=[X_1^{a-1}, X_{a+1}^{2a-1}, ..., X_{(\\gamma-1)a+1}^{\\gamma a-1}, X_{\\gamma a+1}^\\infty]$$\nare conditionally independent of each other. Based on this condition, we have the following seedless construction of\nvariable-length extractors.\n\n\\begin{Construction}\\label{const:4}\nGiven a source $\\mathcal{R}$ described by (\\ref{equ_seedless_1}), we can construct a seedless variable-length extractor $E$ in the following way:\n\\begin{enumerate}\n  \\item Suppose that $$H_{\\min}(X^{(i)}|\\lambda^{(i)}, \\lambda^{(i+1)})\\geq k_{d}, \\forall 0\\leq i\\leq \\gamma.$$\n  We construct a $\\gamma$-source extractor \\cite{Rao2007} $E_1:[\\{0,1\\}^{a-1}]^\\gamma \\rightarrow \\{0,1\\}^d$ such that if each source has\n  min-entropy $k_d$, it can extract $d$ almost-random bits which are $\\epsilon_1$-close to the uniform distribution on $\\{0,1\\}^d$.\n  \\item We construct a seeded variable-length extractor $E_2: S_p\\times \\{0,1\\}^d\\rightarrow\\{0,1\\}^m$ such that\n  with condition on $\\lambda^{(\\gamma)}$,\n  it can extract $m$ almost-random bits from $X^{(\\gamma+1)}$ and these $m$ almost-random bits are $\\epsilon_2$-close to the uniform distribution on\n  $\\{0,1\\}^m$ if the seed is truly random.\n  \\item The seedless variable-length extractor $E$ is a cascade of $E_1$ and $E_2$:\n  Let $$D=E_1(X^{(1)}, X^{(2)},..., X^{(\\gamma)}),$$\n  then we apply $D$ as the seed of $E_2$ to generate $m$ almost-random bits from $X^{(\\gamma+1)}$; that is,\n  $$E(X)=E_2(X^{(\\gamma+1)}, E_1(X^{(1)}, X^{(2)},..., X^{(\\gamma)})).$$\n  \\end{enumerate}\n\\end{Construction}\n\nFor this construction, we have the following theorems.\n\n\\begin{Theorem} In Construction \\ref{const:4}, the $m$ almost-random bits generated by the seedless variable-length extractor $E$ are $(\\epsilon_1+\\epsilon_2)$-close to the uniform\ndistribution on $\\{0,1\\}^m$.\n\\end{Theorem}\n\n\\proof According to the construction, we can let the parameter $a=|X^{(i)}|+1$ with $1\\leq i\\leq \\gamma$ be large enough, so given $\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}$,\n$$\\|D-U_d\\|\\leq \\epsilon_1.$$\n\nLet $X_m$ be the input sequence of $E_2$ that read from $X^{(\\gamma+1)}$, then given  $\\lambda^{(\\gamma)}$,\nwe have\n$$\\|E_2(X_m, U_d)-U_m\\|\\leq \\epsilon_2.$$\n\nFrom the two inequalities above, given $\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}$,\nwe have\n$$\\|E_2(X_m, D)-U_m\\|\\leq \\epsilon_1+\\epsilon_2.$$\n\nSince it is true for any assignments of  $\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}$, we can get\n\\begin{eqnarray*}\n  &&  \\|E_2(X_m, D)-U_m\\|\\\\\n   &=& \\sum_{\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}}P[\\lambda^{(1)}, \\lambda^{(2)}, ..., \\lambda^{(\\gamma)}] (\\epsilon_1+\\epsilon_2)\\\\\n   &\\leq & \\epsilon_1+\\epsilon_2.\n\\end{eqnarray*}\n\nHence, the $m$ almost-random bits extracted by $E$ is also $(\\epsilon_1+\\epsilon_2)$-close to $U_m$.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nIn the following theorem, we show that the seedless variable-length extractor $E$ has the efficiency as\nthe seeded variable-length extractor $E_2$.\n\n\\begin{Theorem} In Construction \\ref{const:4}, suppose that\n$$H_{\\min}(X^{(i)}|\\lambda^{(i)}, \\lambda^{(i+1)})=\\Theta(|X^{(i)}|), \\forall 0\\leq i\\leq \\gamma.$$\nLet $\\eta_{E}$ denote the efficiency of the resulting seedless variable-length extractor $E$, and\nlet $\\eta_{E_2}$ denote the efficiency of the $E_2$, then\n$$\\eta_E=\\eta_{E_2}.$$\n\\end{Theorem}\n\n\\proof According to the construction of $E_1$, we can get that\n$$d=\\Theta(a),$$\nwhere $a=|X^{(i)}|+1$ for $1\\leq i\\leq \\gamma$.\n\nIf $\\epsilon_2$ is a constant, then\n$$d=O(\\log m)=o(m).$$\n\nAs a result, $$\\lim_{m\\rightarrow\\infty}\\frac{a\\gamma}{m}=0.$$\n Let $H$ denote the entropy of the input sequence of $E_2$,\nthen\n$\\eta_{E_2}=\\lim_{m\\rightarrow\\infty}\\frac{m}{H}$, and\n$$\\lim_{m\\rightarrow\\infty}\\frac{m}{a\\gamma + H} \\leq \\eta_E \\leq \\lim_{m\\rightarrow\\infty}\\frac{m}{H}.$$\n\nHence, $\\eta_E=\\eta_{E_2}$.\n\\hfill\\IEEEQED\\vspace{0.1in}\n\nThe theorem above shows that the efficiency of seedless variable-length extractors can be very close to optimality. For many sources, such as biased coins with noise, or Markov chains with noise, the existing algorithms for ideal sources (e.g., perfect biased coins or perfect Markov chains)\ncannot generate high-quality random bits from them. At the same time, the traditional approaches of fixed-length extractors are not very efficient. The gap between their efficiency and the optimality is determined by the bias of the source. Seedless variable-length extractors take the advantages of both, as a result, they can approach the information-theoretic upper bound on efficiency while being capable to combat noise in the sources.\n\n\n\\section{Conclusion and Discussion}\n\nIn this paper, we introduced the concept of the variable-length extractors, namely, those extractors with variable input length and\nfixed output length. Variable-length extractors are generalizations of the existing algorithms for ideal sources to manage general stochastic processes. They are also improvements of traditional fixed-length extractors to fill the gap between min-entropy and entropy of the source\non efficiency. The key idea of constructing variable-length extractors is to approximate the source using a simple model, which is a known process,\na biased coin, or a stationary ergodic process. Depending on the model selected, we proposed and analyzed three seeded constructions\nof variable-length extractors. Their efficiency is lower bounded by $1-\\beta$ and upper bounded by $1$ (optimality), where\n$\\beta (0\\leq \\beta \\leq 1)$ indicate the uncertainty of the real source. We also show that our constructions are asymptotically optimal, in the sense that one cannot find a construction whose efficiency is always strictly larger than $1-\\beta$.\n In addition, we demonstrated how to construct\nseedless variable-length extractors by cascading seeded variable-length extractors with seedless fixed-length extractors. They can work for many (but not all) natural sources such as those based on noise signals.\n\nThere are certain connections between variable-length extractors and a whole family of variable-to-fixed length source codes \\cite{Lawrence1977,Tjalkens1992,Merhav1992,Savari1997,Tjalkens1987,Tunstall1967,Visweswariah2001}. With a variable-to-fixed length code, an infinite sequence is parsed into variable-length phases, chosen from some finite set $\\mathcal{D}$ of phases. Each phase is then\ncoded into a binary sequence of fixed length $m$. The set $\\mathcal{D}$ of phases is complete, i.e., every infinite sequence has a prefix in $\\mathcal{D}$. The key of constructing a good variable-to-fixed length source code is to find the best set $\\mathcal{D}$ that consists of at last $2^m$ prefix-free phases and maximizes the expected phase length. As comparison, the key of constructing a variable-length extractor is to find\nthe best input set $S_p$ that consists of sequences with probability at most $2^{-k}$ for each and minimizes the expected sequence length.\nAlthough their goals are different, some common ideas can be used to construct both the phase set $\\mathcal{D}$ and the input set $S_p$.\nFor example, in \\cite{Visweswariah2001}, Visweswariah et al.  defined  the phase set $\\mathcal{D}$ by $x^*\\in \\mathcal{D}$ if $P(x^*)\\leq c$ and no prefix of $x^*$ satisfies this property. The same idea is applied in our construction I.  In \\cite{Lawrence1977,Tjalkens1992},\nthe phase set $\\mathcal{D}$ is determined by the number of ones and zeros in the phase, so is our construction II.  In some sense,\nan optimal variable-to-fixed length code can result in a fixed-length binary sequence whose min-entropy is close to its length. However,\nvariable-to-fixed length source codes do not always work well in constructing variable-length extractors, because\n(1) the designing criteria are different and they may degrade the performance; (2) variable-to-fixed length source codes take both encoding and decoding in consideration, hence, they are more complex in computation than what we require (decoding is not necessary) for constructing variable-length extractors; and (3) the sources that we considered for variable-length extractors are unpredictable, which are more general than the ones considered in variable-to-fixed length source codes.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}