Rational double points on supersingular K3 surfaces
Ichiro Shimada
研究了超奇异$K3$曲面上总Milnor数为21的有理二重点的构型.给出了可能配置的完整列表。作为应用，我们还给出了超奇异$K3$曲面上极值（准）椭圆纤维化的完整列表。
We investigate configurations of rational double points with the total Milnor number 21 on supersingular $K3$ surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular $K3$ surfaces.
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In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.
Rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3
本文用多种方法证明了特征3中Artin不变量为1的超奇异K3曲面上112条非奇异有理曲线的存在性。利用这些有理曲线，我们得到了K3曲面上的$(16)_{10}$-构型和$(280_{4}, 112_{10})$-构型.此外，我们还利用Leech格的理论研究了Picard格。112条非奇异有理曲线对应于112个Leech根。
Supersingular K3 surfaces in odd characteristic and sextic double planes
证明了超奇异K3曲面对射影平面的双重覆盖是双有理的。
Supersingular K3 surfaces in characteristic 2 as double covers of a projective plane
对于特征2中的超奇异$K3$曲面$X$，存在6次齐次多项式$G$，使得$X$对于由$w^2=G$定义的射影平面的纯不可分双覆盖是双有理的。我们提出了一个算法，从$G$计算一组生成元的数值Néron-Severi格的$X$。作为应用，我们研究了特征为2的2次超奇异$K3$曲面的模空间上的Artin不变量所定义的分层。
A note on a supersingular K3 surface in characteristic 2
在特征为2且Artin不变量为1的超奇异K3曲面上，利用广义Kummer曲面的结构构造了一组21条不相交的光滑有理曲线和一组21条不相交的光滑有理曲线，使得其中一组曲线中的每条曲线与另一组曲线中的5条重数为1的曲线恰好相交.
On the supersingular K3 surface in characteristic 5 with Artin invariant 1
我们提出了三个有趣的投影模型的超奇异K3表面X的特征5与阿廷不变量1。对于每一个射影模型，我们确定光滑的有理曲线在X上的最小程度和射影自同构群。利用超特殊交换曲面构造了X上6组16条不相交的光滑有理曲线，并证明了它们构成了一个美丽的构形。
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…
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 …
Rational curves in the moduli of supersingular K3 surfaces
我们展示了如何使用Artin-Tate同构的相对形式和Bridgeland在椭圆曲面上稳定层的模空间上的结果的扭曲类似物来构造有理曲线上的超奇异K3曲面的非等平凡族。因此，我们表明，每一个点的阿丁不变10标记超奇异K3表面的Ogus空间在于无限多对不同的有理曲线规范相关的椭圆结构的基础K3表面。
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。
The Number of Rational Points On Genus 4 Hyperelliptic Supersingular Curves in Characteristic 2
有限域上曲线领域的一个大问题是关于点的个数的分布：在亏格为$g$的曲线上，哪些数是点的个数？同样的问题也可以问到曲线的各种子类。本文对$2^n$，$n$奇阶有限域上亏格为4的超奇异超椭圆曲线上点数的可能性进行了分类.
A 1-dimensional family of Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1
我们给出了一个1维族的经典和超奇异Enriques曲面的特征2覆盖的超奇异K3表面与Artin不变1。证明了在Neron-Severi格模挠的正交群中，反射群为有限指数的Enriques曲面上存在30条非奇异有理曲线和10个非有效（-2）-因子.
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。
On Ballico-Hefez curves and associated supersingular surfaces
设p是素数，q是p的幂，Ballico-Hefez曲线是特征p为q+1次的非自反结点有理平面曲线，本文研究了它的自同构群和定义方程.我们还证明了作为沿Ballico-Hefez曲线分支的射影平面的循环覆盖所得到的曲面是单有理的，因此是超奇异的。作为应用，我们得到了特征3和特征5中Artin不变量为1的超奇异K3曲面的一个新的投影模型。
Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5
设X是特征为5且Artin不变量为1的超奇异K3曲面。那么X有一个极化，它把X实现为费马六次双平面。给出了X的一系列与该Fermat六次极化的相交数小于小于或等于5的2次极化，并给出了相应的投影模型的定义方程.我们还提出了一种方法来描述这些投影模型之间的双有理态射显式。作为副产品，一个非投射自同构的费马六次双平面得到。
Dynkin diagrams of rank 20 on supersingular K3 surfaces
我们对特征p的Milnor数为20的正规超奇异K3曲面Y进行分类，其中p是不整除Y上有理二重点的Dynkin型判别式的奇素数.
A supersingular K3 surface in characteristic 2 and the Leech lattice
我们在特征为2的代数闭域上构造了一个K3曲面，该曲面包含两组21条不相交的光滑有理曲线，使得一组曲线中的每条曲线与另一组曲线中的5条曲线恰好相交.这种构形同构于4元有限域上的射影平面上的点和线的构形。该表面承认一个有限的自同构群，其中包含的自同构群的平面上的每一组21个光滑的有理曲线的配置上的行为，和额外的元素，为了交换两套。该表面的Picard格是Signatuire（1，25）的偶么模格的反射子格，并且42条曲线的类对应于该格中的一些Leech根。
On normal K3 surfaces
确定了复正规代数K3曲面和特征p > 19的正规超奇异K3曲面上有理二重点的所有可能构型。
Planar, rational curves over ${\mathbb F}_2$ whose only singularity is a double point
研究了 over ${\mathbb F}_2$ 的平面有理曲线，其唯一奇点为二重点，且在特征0时仅限于度数不超过6的情况。
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 singular property of the supersingular elliptic curve in characteristic 2
设E是定义在k上的超奇椭圆曲线，它是二元有限域的代数闭包，在k-同构下是唯一的.用0表示它的单位元，设C是E-{0}在群Isom（E）（它是非阿贝尔的，阶数为24）作用下的商。本文的主要结果是：集合C（k）自然地参数化Lamé覆盖的k-同构类，Lamé覆盖是在具有特定分歧基准的三点之外未分歧的射影线的平凡分歧覆盖.这一事实是令人惊讶的原因有两个：首先，它是第一个非平凡的例子，一个家庭的覆盖的射影线unramified以外的三个点，这是参数化的几何点的曲线。此外，当考虑任意特征时，Lamé覆盖的显式构造非常复杂，并且它们的算术性质仍然神秘。当结合从正特性到特性0的提升技术时，特性2中的问题的简单性具有许多深刻的后果。作为一个例子，我们得到了一些关于（局部）伽罗瓦作用的尖锐陈述，例如，在一个固定数域上定义的Lamé覆盖的同构类的数量上的界，仅取决于在2处的剩余扩张的程度。最后，在附录中我们给出了这些结果的部分推广，证明了对任意正整数g，亏格g超椭圆曲线的适当商的k-有理点点参数化了在三点外未分歧的射影线的tamely分歧覆盖的k-同构类。
Kummer surfaces associated with group schemes
本研究在特征2情形下引入由群概形mu_2作用于有理尖点曲线自乘积得到的Kummer曲面，证明其恰为Artin不变量σ≤3的超奇异K3曲面，给出曲线构型刻画并说明其可作为单连通Enriques曲面的正规K3型覆盖。
We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.
The Dynkin diagrams of rational double points
有理二重点是最简单的曲面奇点。本文主要研究有理二重点分解所对应的例外集的几何性质。我们将根据Dynkin图导出有理二重点的分类。
We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.
The Hilbert schemes of points on surfaces with rational double point singularities
证明了在最坏情况下具有有理二重点奇点的正规拟射影曲面上的点的Hilbert格式是不可约的。
A very general sextic double solid is not stably rational
我们证明了沿着非常一般的六次曲面分支的P^3的双重覆盖不是稳定有理的。
Potential density of rational points on sextic double solids
本文构造了特征零域上光滑六次双固体的例子，证明其有理点集合具有潜在稠密性。
We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.
Arcs on rational double points in arbitrary characteristic
该研究证明纳什问题在所有特征下对二维有理二重点成立，核心方法是直接计算通过这些奇点的弧族。
Painlevé equations and deformations of rational surfaces with rational double points
本文证明了Painlevé方程的Bäcklund变换来自于Okamoto构造的有理曲面作为初始条件空间的双有理映射.有理曲面族的有理二重点的同时分解产生了触发器，触发器是Painlevé方程对称群的起源。同时指出Kodaira-Spencer有理曲面变形理论解释了Painlevé方程的几何意义。
A very general quartic double fourfold or fivefold is not stably rational
应用C. Voisin，我们证明了沿着非常一般的四次超曲面分支的P^4或P^5的双覆盖不是稳定有理的。
In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo surface with rational double points whose anti-canonical members are all singular. As a corollary, we determine all singularity types of del Pezzo surfaces with rational double points which only appear in positive characteristic.
Non-Cyclic Subgroups of Jacobians of Genus Two Curves
设E是定义在有限域上的椭圆曲线。Balasubramanian和Koblitz证明了：如果单位m_l的l次根不包含在基域中，则基域的一个域扩张包含m_l当且仅当E的l-扭点在同一个域扩张上有理。我们将这个结果推广到亏格为两条曲线的Jacobian。特别地，我们证明了在基域的相同域扩张上，Weil对和Tate对是非简并的.由此得到了超奇异亏格两条曲线的Jacobian的l-扭子群的完整刻画。特别地，我们证明了当l>3时，l-挠点在次数至多为24的域扩张上是有理的。
On rational double points over nonclosed fields
本文计算了有理双点奇点的方程，并确定了其在完美基域上作为SL_{2,k}有限线性约化子群方案商奇点的类型。
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。
We study a double solid X branched along a nodal sextic surface in a projective space and the 2-torsion subgroup in the third integer cohomology group of a resolution of singularities of X. This group can be considered as an obstruction to rationality of X. Studying this group we conclude that all sextic double solids admitting non-trivial obstructions to rationality are branched along determinantal surfaces of very specific type and we provide an explicit list of them.
Jet schemes and minimal toric embedded resolutions of rational double point singularities
利用有理二重点奇点的喷流格式的结构，我们构造了这些奇点的“最小嵌入环面分解”。我们还建立，这些奇点，一个自然类的不可约组件的喷气计划集中在奇异轨迹和一组因子出现在每个“最小嵌入复曲面决议”之间的对应关系。我们证明了除了$E_8$的奇异性之外，这个对应是双射的.这可以看作是有理二重点奇点的嵌入纳什对应。
Which rational double points occur on del Pezzo surfaces?
本研究确定了任意特征代数闭域上任意次数与皮卡秩的RDP del Pezzo曲面的所有有理二重点构型，推广了Du Val的经典结果，并给出了含非紧有理二重点的1次RDP del Pezzo曲面的简化方程。
Rationally connected rational double covers of primitive Fano varieties
本研究证明M≥5时P^{M+1}中M+1次Zariski一般超曲面不存在阿贝尔群的高次Galois有理连通覆盖，尤其不存在2次有理连通有理映射，该结论可推广至其他本原Fano簇并启发绝对刚性猜想。
Moduli of K3 surfaces of degree 2 with four rational double points of type $D_4$
该研究确定了带四个$D_4$型有理二重点的2次格极化K3曲面模空间的Satake-Baily-Borel紧化为三维射影空间，并刻画了其自守形式分次环的生成元与关系。
Perfectoid pure thresholds of lifts of rational double points
本文研究了在混合特征(0,p)下，与p相关的完美化纯阈值，该阈值是奇异性的不变量。通过计算理性质点提升的完美化纯阈值，证明了这些阈值集合包含在有理数中，并满足升链条件。在特征2的情况下，所有正整数的倒数出现，0是唯一的极限点。
On the projective normality of double coverings over a rational surface
本文研究了极小曲面$X$的投影正规性，它是有理曲面$X$上的分歧双重覆盖，具有$\dim|-K_S|\ge 1$。特别是Horikawa曲面，一般类型的极小曲面与$K^2_X=2p_g(X)-4$，是这种类型，直到奇点的分解。令$π$为从$X$到$S$的覆盖映射。证明了$\mathbb{Z}_2$ -不变伴随因子$K_X+rπ^*A$是正规生成的，其中整数$r\ge 3$和$A$是$S$上的一个充分因子.
Varieties of minimal rational tangents on double covers of projective space
设$ϕ: X \to \mathbb P^n$是一个沿光滑超曲面分支的双复盖，超曲面的次数为$2m, 2 \leq m \leq n-1$ .研究了在$X$的一般点$x$上的极小有理切线$\mathcal C_x \subset \mathbb P T_x(X)$的簇.我们描述了$\mathcal C_x$的齐次理想，并证明了当$x$在$X$的一般点上变化时，$\mathcal C_x$的投射同构型以极大的方式变化.我们对$\mathbb C_x$的理想的描述暗示了覆盖态射$ϕ$的某种刚性性质。作为这个刚性的应用，我们证明了任何有限态射之间的这种双重覆盖$m=n-1$必须是同构。证明了$X$上极小有理曲线的Liouville型扩张性质成立.
Doubling rational normal curves
本文研究了有理正态曲线上的二重结构。在回顾\cite{fer}中描述的光滑曲线上支撑的二重结构的一般构造后，我们将其专门化为有理正态曲线上的二重结构。对于每一个双曲线结构，我们关联一个整数三元组$ (2r,g,n) $，其中$ r $是支撑度，$ n \geq r $是包含双曲线的射影空间的维数，$ g $是双曲线的算术亏格。我们还计算一些数值不变量的构造曲线，我们表明，家庭的双重结构与一个给定的三重$ (2r,g,n) $是不可约的。进一步证明了$ (2r,r+1,r) $和$ (2r,1,2r-1) $族中的一般双曲线是算术Gorenstein曲线.最后证明了含亏格为$ g \leq -2 $的双二次曲线的轨迹的闭包构成相应Hilbert格式的不可约分量，且一般双二次曲线是该分量的光滑点.证明了任意亏格的$ \mathbb{P}^3 $中的一般双二次曲线是相应Hilbert格式的光滑点。
Srinivas' Problem for Rational Double Points
对于局部几何正规整环的完备化B，V. Srinivas提出了一个问题：当A在正规几何整环之间变化，而B同构于A的完备化时，Cl B的哪些子群作为从Cl A到Cl B的映射在类群上的像而出现。对于二维有理二重点奇点，我们证明了所有子群都是这样产生的。我们还表明，在任何维，每个正常的超曲面奇点有完成同构的几何UFD。我们的方法是全球性的，应用Noether-Lefschetz理论的线性系与非约化的基础轨迹。
On the classification of surfaces of general type with non-birational bicanonical map and Du Val Double planes
我们对一般类型的曲面进行了分类，其双正则映射是由一个2次有理映射到有理曲面或直纹曲面上的。
Rational elliptic surfaces with six singular double fibres
本文研究具有六个奇异双纤维的有理椭圆面，通过多种视角进行分类并给出构造方法。
Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over …
Simple surface singularities
由著名的ADE分类理性二重点是简单的。有理三重点也很简单。我们猜想简单法曲面奇点就是有理奇点，其分解图可以由有理二重点或有理三重点的图通过使（某些）顶点权值更负得到.对于有理奇点，我们通常展示一个方向，而在有理四重点和夹层奇点的特殊类别中展示另一个方向（简单性）。
Threefold Flops via Matrix Factorization
由冈萨雷斯-斯普林伯格和维迪埃提出的显式麦凯对应关系，将有理二重点的最小分解中的每个例外因子与有理二重点方程的矩阵分解相关联。我们研究了这些矩阵分解的变形，并证明了它们存在于A型和D型有理二重点的适当的“部分分解”变形空间上。因此，所有长度为1和2的简单触发器都可以用矩阵分解定义的爆破来描述。我们还制定将这些结果推广到合理的二重点的E型和简单的触发器的长度大于2。
Double ramification loci, also known as strata of $0$-differentials, are algebraic subvarieties of the moduli space of smooth curves parametrizing Riemann surfaces such that there exists a rational function with prescribed ramification over $0$ and $\infty$. We describe the closure of double ramification loci inside the Deligne-Mumford compactification in geometric terms. To a rational function we associate its exact differential, which allows us to realize double ramification loci as linear subvarieties of strata of meromorphic differentials. We then obtain a geometric description of the clos…
A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. Looijenga proved in 1981 that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. In 1983, the second author and Miranda gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K-trivial semistable model for the central fiber of such a smoothing. We study these "Type III degenerations" of rational surf…
Double Rational Normal Curves with Linear Syzygies
一种是描述有理正态曲线上的二重结构，这种曲线是由二次方程从理论上定义的，具有线性合点，将平面上的二重线推广到平面上
To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$ over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jets on $X$. Its fiber over a singular point of $X$ is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of $X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg 0$. In this paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-type, we study the intersections of irreducible compone…
Gröbner Fans and Minimal Embedded Toric Resolutions of Rational Double Point Singularities
本文通过显式构造ADE奇点的Gröbner扇并利用profile概念进行加细，重构了有理二重点奇点的嵌入环面消解。
A very general quartic double fourfold is not stably rational
我们证明了一个非常一般的双重覆盖的投射四空间，分歧在四次三倍，是不稳定的理性。
We classify purely inseparable morphisms of degree $p$ between rational double points (RDPs) in characteristic $p > 0$. Using such morphisms, we refine a result of Artin that any RDP admits a finite smooth covering.
We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.
The 2-divisibility of divisors on K3 surfaces in characteristic 2
本文研究了特征2下K3曲面上可被2整除的有理曲线组，证明了n=8,12,16,20均可在超奇异K3曲面上实现，而有限高度情形仅n=8成立。
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