Kai Behrens
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.
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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 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…
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.
We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number $h^{2,0} > 1$, even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the hodge number formula for Göttsche-Soergel does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showin…
An Enriques involution of a supersingular K3 surface over odd characteristic
本文利用Crystalline Torelli定理证明了在复情形下，奇特征域上的超奇异K3曲面具有Enriques对合当且仅当存在Enriques格的二次到Neron-Severi群的本原嵌入，使得嵌入的正交补没有自交向量-2.利用这一判据和一些格点计算，我们证明了当基场的特征为p=19或p>23时，超奇异K3曲面是Enriques K3曲面当且仅当Artin不变量小于6.
Let $\bar{Y}$ be a normal surface that is the canonical $μ_2$- or $α_2$-covering of a classical or supersingular Enriques surface in characteristic $2$. We determine all possible configurations of singularities on $\bar{Y}$, and for each configuration we describe which type of Enriques surfaces (classical or supersingular) appear as quotients of $\bar{Y}$.
On Enriques surfaces in characteristic 2 with a finite group of automorphisms
将具有有限群的自同构的复Enriques曲面分为七类。在本文中，我们确定哪些类型的这样的Enriques曲面存在于特征2。特别地，我们给出了一维经典和超奇异Enriques曲面族，其自同构群同构于五次对称群。
We classify supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 8 types according to their dual graphs of all $(-2)$-curves (nonsigular rational curves). We give examples of these Enriques surfaces together with their canonical coverings. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished.
We give a complete description of all classical Enriques surfaces with non-zero global vector fields. In particular we show that there are such surfaces. The obtained result also applies to supersingular Enriques surfaces fulfilling a rather special condition. During this classification we study some properties of genus 1-fibrations special to characteristic as well as make a close study of the genus 1-fibration on the surfaces that we classify.
An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second $l$-adic cohomology group (this group modulo torsion subgroup). Extending the results of S. Mukai and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $…
A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2
本文研究特征2下ℙ²×ℙ²上的秩2不可分解向量丛E的零集，构造出双次数为(4,4)的非经典Enriques曲面族，并证明双线性正规的此类曲面均为此类零集。
Gaussian maps for singular curves on Enriques surfaces
本研究利用高斯映射给出带标记Prym曲线存在满足特定奇点条件的Enriques曲面奇异模型的阻碍条件。
Non-classical Godeaux Surfaces
非经典Godeaux曲面是一般类型的极小曲面，它具有$χ=K^2=1$但具有$h^{01}\neq0$。我们证明了这样的曲面满足$h^{01}=1$，并且它们只能存在于正特征不超过5的域上。像非经典的恩里克曲面，他们分为两类：奇异和超奇异的。给出了它们在特征5中的完全分类，并计算了它们的Hodge-、Hodge-Witt-和结晶上同调（包括挠率）.最后，我们给出了一个超奇异的戈多曲面的特征5的例子。
The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the Néron-Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point free involution that gives r…
Explicit birational models of marked elliptic surfaces of relative degree two
本文针对带双截面或二次相对极化、含二重纤维的椭圆曲面，提出两种双有理模型的显式构造方法，并将其简洁描述应用于指数2的Halphen曲面与Enriques曲面。
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。
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…
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 give several results concerning the connected component ${\rm Aut}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behaviour with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces $f: X \to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ of the space of global vector fields…
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