Ichiro Shimada
We construct explicitly moduli curves of polarized supersingular K3 surfaces in characteristic 2 with Artin invariant 2. As an application, we detect a "jump" phenomenon in a family of automorphism groups of supersingular K3 surfaces with a constant Néron-Severi lattice.
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We prove that the supersingular K3 surface of Artin invariant 1 in characteristic p (where p denotes an arbitrary prime) admits a model over IF_p with Picard number 21.
We classifiy Enriques surfaces covered by the supersingular K3 surface with the Artin invariant 1 in characteristic 2. There are exactly three types of such Enriques surfaces.
We describe all the elliptic models with section on the Shioda supersingular K3 surface X of Artin invariant 1 over an algebraically closed field of characteristic 3. In particular, we compute elliptic parameters and Weierstrass equations for the fifty two different fibrations, and analyze the reducible fibers and Mordell-Weil lattices.
We give a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1. Moreover we show that there exist 30 nonsingular rational curves and ten non-effective (-2)-divisors on these Enriques surfaces whose reflection group is of finite index in the orthogonal group of the Neron-Severi lattice modulo torsion.
We give a short proof that every supersingular K3 surface (except possibly in characteristic $2$ with Artin invariant $σ=10$) has an automorphism of Salem degree 22. In particular an infinite subgroup of the automorphism group does not lift to characteristic zero. The proof relies on the case $σ=1$ and the cone conjecture for K3 surfaces.
In characteristic $p=0$ or $p>5$, we show that a K3 surface with an order 60 automorphism is unique up to isomorphism. As a consequence, we characterize the supersingular K3 surface with Artin invariant 1 in characteristic $p=11$ (mod 12) by a cyclic symmetry of order 60.
The automorphism group of a supersingular K3 surface with Artin invariant 1 in characteristic 3
该研究针对特征3下阿廷不变量为1的超奇异K3曲面，给出其自同构群的有限生成元集合。
We present three interesting projective models of the supersingular K3 surface X in characteristic 5 with Artin invariant 1. For each projective model, we determine smooth rational curves on X with the minimal degree and the projective automorphism group. Moreover, by using the superspecial abelian surface, we construct six sets of 16 disjoint smooth rational curves on X, and show that they form a beautiful configuration.
We show the existence of 112 non-singular rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3 by several ways. Using these rational curves, we have a $(16)_{10}$-configuration and a $(280_{4}, 112_{10})$-configuration on the K3 surface. Moreover we study the Picard lattice by using the theory of the Leech lattice. The 112 non-singular rational curves correspond to 112 Leech roots.
The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to classify the fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (-2)-curves on X related to the incidence graph of points and lines of IP^2(IF_4).
Let X be a supersingular K3 surface in characteristic 5 with Artin invariant 1. Then X has a polarization that realizes X as the Fermat sextic double plane. We present a list of polarizations of X with degree 2 whose intersection number with this Fermat sextic polarization is less than or equal to 5, and give the defining equations of the corresponding projective models. We also present a method to describe birational morphisms between these projective models explicitly. As a by-product, a non-projective automorphism of the Fermat sextic double plane is obtained.
We show that the Mathieu groups $M_{22}$ and $M_{11}$ can act on the supersingular $K3$ surface with Artin invariant 1 in characteristic 11 as symplectic automorphisms. More generally we show that all maximal subgroups of the Mathieu group $M_{23}$ with three orbits on 24 letters act on a supersingular $K3$ surface with Artin invariant 1 in a suitable characteristic.
Let X and Y be supersingular K3 surfaces defined over an algebraically closed field. Suppose that the sum of their Artin invariants is 11. Then there exists a certain duality between their Néron-Severi lattices. We investigate geometric consequences of this duality. As an application, we classify genus one fibrations on supersingular K3 surfaces with Artin invariant 10 in characteristic 2 and 3, and give a set of generators of the automorphism group of the nef cone of these supersingular K3 surfaces. The difference between the automorphism group of a supersingular K3 surface X and the automorp…
In this short paper we prove that the following two 1-dimensional families of Artin-Schreier curves are supersingular: y^7 - y = x^5 + c.x^2 over F_7 y^5 - y = x^7 + c.x over F_5 (for some parameter c). Our method is developed upon the p-adic Dwork's method.
Supersingular Tate conjecture for irreducible symplectic varieties of known types: I
本文证明多种已知类型的超奇异不可约辛簇在特定Artin不变量条件下具有Tate或超奇异阿贝尔Chow motive，进而其乘积满足超奇异Tate猜想。
We show that every supersingular K3 surface in characteristic 5 with Artin invariant less than or equal to 3 is unirational.
We show that most classes of K3 surfaces have only finitely many Enriques quotients. For supersingular K3 surfaces over fields of characteristic $p \geq 3$, we give a formula which generically yields the number of their Enriques quotients. We reprove via a lattice theoretic argument that supersingular K3 surfaces always have an Enriques quotient over fields of small characteristic. For some small characteristics and some Artin invariants, we explicitly compute lower bounds for the number of Enriques quotients of a supersingular K3 surface. We show that the supersingular K3 surface of Artin inv…
We construct a moduli space of adequately marked Enriques surfaces that have a supersingular K3 cover over fields of characteristic $p \geq 3$. We show that this moduli space exists as a scheme locally of finite type over $\mathbb{F}_p$. Moreover, there exists a period map from this moduli space to a period scheme and we obtain a Torelli theorem for supersingular Enriques surfaces.
We show that every supersingular K3 surface in characteristic 2 with Artin invariant less than or equal to 2 is obtained by the Kummer type construction of Schroeer.
In this note we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes p congruent 3 mod 4 in a systematic way. Automorphisms of Salem degree 22 do not lift to any characteristic zero model.
We present a method to generate many automorphisms of a supersingular K3 surface in odd characteristic. As an application, we show that, if p is an odd prime less than or equal to 7919, then every supersingular K3 surface in characteristic p has an automorphism whose characteristic polynomial on the Néron-Severi lattice is a Salem polynomial of degree 22. For a supersingular K3 surface with Artin invariant 10, the same holds for odd primes less than or equal to 17389.
We show how to construct non-isotrivial families of supersingular K3 surfaces over rational curves using a relative form of the Artin-Tate isomorphism and twisted analogues of Bridgeland's results on moduli spaces of stable sheaves on elliptic surfaces. As a consequence, we show that every point of Artin invariant 10 in the Ogus space of marked supersingular K3 surfaces lies on infinitely many pairwise distinct rational curves canonically associated to elliptic structures on the underlying K3 surface.
We construct, on a supersingular K3 surface with Artin invariant 1 in characteristic 2, a set of 21 disjoint smooth rational curves and another set of 21 disjoint smooth rational curves such that each curve in one set intersects exactly 5 curves from the other set with multiplicity 1 by using the structure of a generalized Kummer surface.
Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q+1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.
For every supersingular $K3$ surface $X$ in characteristic 2, there exists a homogeneous polynomial $G$ of degree 6 such that $X$ is birational to the purely inseparable double cover of a projective plane defined by $w^2=G$. We present an algorithm to calculate from $G$ a set of generators of the numerical Néron-Severi lattice of $X$. As an application, we investigate the stratification defined by the Artin invariant on a moduli space of supersingular $K3$ surfaces of degree 2 in characteristic 2.
O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supers…
Finite symplectic automorphism groups of supersingular K3 surfaces
本文基于Dolgachev和Keum的工作，对特征p>11的K3表面的有限辛自同构群进行了完整分类，揭示了超奇异K3表面的辛自同构群结构。
Oort's conjecture on automorphisms of generic supersingular abelian varieties
本文证明了Oort猜想：在特征p的亏格g主极化阿贝尔簇模空间的超奇异轨迹上，除g=2或3且p=2的情况外，通用主极化阿贝尔簇的自同构群仅为±1。
Supersingular elliptic surfaces and Infinitesimal Torelli
本文通过产品-商结构证明Katsura的例子是Artin奇点椭圆面，并给出新的证明表明自身例子不满足微分Torelli条件。
We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
The Hilbert scheme of points for supersingular abelian surfaces
该研究分析正特征下阿贝尔曲面点Hilbert概型与Beauville广义Kummer簇的几何，证明特征2时两点Hilbert概型的加法映射为拟纤维化且纤维非光滑，揭示奇点与p秩、a数的关联，并建立Artin野对合的McKay对应。
We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted sheaves on universal gerbes associated to the Brauer groups of supersingular K3 surfaces. In rank 0, this is a geometric incarnation of the Artin-Tate isomorphism. Twistor spaces give rise to curves in moduli spaces of twisted supersingular K3 surfaces, analogous to the analytic moduli space of marked K3 surfaces. We describe a theory of crystals for twisted supersingul…
We define Reichstein transforms to be certain birational transformations of Artin stacks with good moduli spaces. Our main technical result is that the Reichstein transform of an Artin toric stack is again an Artin toric stack. This leads to our main theorem which states that for Artin toric stacks there is a canonical sequence of Reichstein transforms which produces a Deligne-Mumford toric stack. When the good moduli space of an Artin toric stack is projective our procedure can be interpreted in terms of Kirwan's partial desingularization of geometric invariant theory quotients.
We prove Artin's axioms satisfy a compatibility for composition of 1-morphisms of stacks in groupoids. Consequently, some natural stacks in groupoids are algebraic, including a common generalization of Vistoli's Hilbert stack and the stack of branchvarieties of Alexeev and Knutson.
We classify elliptic K3 surfaces in characteristic $p$ with $p^n$-torsion sections. For $p^n\geq3$ we verify conjectures of Artin and Shioda, compute the heights of their formal Brauer groups, as well as Artin invariants and Mordell--Weil groups in the supersingular cases.
Moduli spaces of rational curves on Artin-Mumford double solids
研究阿廷 - 芒福德双固体上有理曲线模空间的不可约分量，通过此研究为满足几何马尼恩猜想的法诺簇提供首个例子，有多个马尼恩分量。
In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of $\mathbb F_{p^n}$-rational points on $C_k^{(p,a)}$ for all $n$, and uses a result we proved elsewhere about the number of rational points on supersingular curves.
A Computational Framework for the Moduli of Supersingular Abelian Varieties
建立极化旗型商的结构定理，证明顶层精细正规形和拟极化的可计算下降准则，完成旗型第一和最后一步分类，并计算 $g\leq 5$ 维的极化旗型商。
The paper studies the supersingular locus of the characteristic p moduli space of principally polarized abelian 8-folds that are equipped with an action of a maximal order in a quaternion algebra, that is non-split at the infinite place, but split at p. The main result is that its irreducible components are Fermat surfaces of degree p+1.
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.
Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p \geq 5. We prove Artin's conjecture under the additional assumption that X has a polarization of degree 2d with p > 2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime-to-p when p \geq 5. The argument uses Borcherds' con…
On the non-Transversality of the Hyperelliptic Locus and the Supersingular Locus for $g=3$
本文研究亏格3时西格尔模叠中超椭圆轨迹与超奇异轨迹的交点性质，给出非横截交点的判别准则，并证明对无穷多个素数p存在此类交点。
Algebraicity of supermoduli of curves via Artin's criteria
本文通过超几何的Artin代数性准则证明了四个超几何模群的代数性，包括超曲线、超黎曼面及其稳定形式，其中稳定超黎曼面的模群代数性有新证明，而稳定超曲线的模群代数性为新结果。
For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.
The cycle class of the supersingular locus of principally polarized abelian varieties
本文证明了特征p下主极化阿贝尔簇模空间周环中超奇异轨迹的环类公式，将其表示为霍奇丛陈类的单项式乘以p的多项式因子，并确定了亏格g=4时的因子。
Let $X$ be a smooth proper genus 2 curve over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the the moduli space $M\_X$ of semi-stable rank 2 vector bundles over $X$, which is isomorphic to a 3-dimensional projective space. Y. Laszlo and C. Pauly recently gave the equations of $F$ for an ordinary $X$. Using deformation, we give these equations for a supersingular $X$ and draw some consequences such as the base locus of $F$ (one point), or the stability of the complementary Zariski open set.
In this paper, we prove, as the complex case, a supersingular K3 surface over a field of odd characteristic has an Enriques involution if and only if there exists a primitive embedding of the twice of the Enriques lattice into the Neron-Severi group such that the orthogonal complement of the embedding has no vector of self-intersection -2 using the Crystalline Torelli theorem. By this criterion and some lattice calculation, we prove that when the characteristic of the base field is p=19 or p>23, a superingular K3 surface is an Enriques K3 surface if and only if the Artin invariant is less than…
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.
Let k be a perfect field of characteristic p > 2. In this note, we show that the local moduli space of a non-supersingular K3 surface over k with trivial deformation of the associated enlarged formal Brauer group admits a natural p-divisible formal group structure.
Symmetric Differentials on K3 Surfaces
该研究刻画了代数闭域上K3曲面存在非零正次数整体对称微分的充要条件，推广了Jang定理，明确了特征2超奇异K3曲面的相关结构性质。
Kummer surfaces for the selfproduct of the cuspidal rational curve
针对特征2下经典Kummer构造对超奇异阿贝尔曲面失效的问题，本文用有理尖点曲线自乘积和无穷小群概型作用给出新的Kummer型构造，得到Artin不变量为1、2的超奇异K3曲面，它们属于纯不可分基变换后同步消解得到的1维族。
We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of etale charts admitting good moduli spaces.
We study the good reduction modulo p of K3 surfaces with complex multiplication. If a K3 surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for K3 surfaces with Picard number 20. Our methods rely on the main theorem of complex multiplication for K3 surfaces by Rizov, an explicit description of the Breuil-Kisin modules associated with Lubin-Tate characte…
This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.
We present a complete generalization of Kirwan's partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if $\mathcal{X}$ is an irreducible Artin stack with stable good moduli space $\mathcal{X} \to X$, then there is a canonical sequence of birational morphisms of Artin stacks $\mathcal{X}_n \to \mathcal{X}_{n-1} \to \ldots \to \mathcal{X}_0 = \mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of $\mathcal{X}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of $\mathcal{X}_k$ and the final s…
Using elliptic structures, we show that any supersingular K3 surface of Artin invariant $1$ in characteristic $p \not= 5$, $7$, $13$ has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine different approaches such as quotients by the group scheme $α_p$, Kummer surfaces, and automorphisms of elliptic and hyperelliptic curves.
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus' crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.
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