Arithmetic Moduli and Lifting of Enriques Surfaces

The Deligne-Illusie Theorem and exceptional Enriques surfaces

Building on the results of Deligne and Illusie on liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only the dimension of certain cohomology groups of vector bundles arising from the Frobenius pushforward of the de Rham complex. Using vector bundle methods, we apply this to show that exceptional Enriques surfaces, a class introduced by Ekedahl and Shepherd-Barron, do not lift to truncated Witt vectors, yet the base of the miniversal formal deformation over the Witt vectors is regular. Using the classification of Bombieri and Mumford, we also show that bi…

极化Enriques曲面的模空间——计算方面

Moduli of polarized Enriques surfaces -- computational aspects

本文研究了极化Enriques曲面模空间的算术群，证明共轭意义下恰有87个此类群，且所有模空间均由度为1240的模空间统治，并计算了相关群的Tits building。

Algebraic Surfaces in Positive Characteristic

These notes are an introduction to and an overview of the theory of algebraic surfaces over algebraically closed fields of positive characteristic. After some background in characteristic-p-geometry, we sketch the Kodaira-Enriques classification. Next, we turn to more special characteristic-p topics, and end by lifting results and applications to characteristic zero. We assume that the reader has a background in complex geometry and has seen the Kodaira-Enriques classification of complex surfaces before.

The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type

Consider a cubic surface satisfying the mild condition that it may be described in Sylvester's pentahedral form. There is a well-known Enriques or Coble surface S with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and the (-2)-curves of S. In the case of pentahedral parameters (1, 1, 1, 1, nonzero t) we compute the automorphism group of S. For t not 1 it is the semidirect product of the free product (Z/2)*(Z/2)*(Z/2)*(Z/2) by the symmetric group S4. In the special case t=1/16 we study the action of Aut(S) on an invariant smooth rational…

算术 Kodaira–Spencer 类与通过 Frobenius–Witt 余切复形的 Frobenius 提升

Arithmetic Kodaira--Spencer Class and Frobenius Liftings via Frobenius--Witt Cotangent Complex

本文针对平坦 $\bbZ_{(p)}$-概型，引入基于 Frobenius–Witt 余切复形的算术 Kodaira–Spencer 类，证明其与经典形变理论的 Frobenius 提升障碍等价，还定义了兼容基提升的相对障碍类。

高阶休眠算符的算术提升与二维TQFT

Arithmetic liftings and 2d TQFT for dormant opers of higher level

本文研究正特征下休眠算符的模空间，构造了紧化模空间，证明了n=2时的平展性并导出典范对角提升，还发现次数函数构成2d TQFT，提供了计数这类算符和二阶微分方程的有效方法。

Periods of an arrangement of six lines and Campedelli surfaces

We define a period map for classical Campedelli surfaces, using a covering trick as in the case of Enriques surfaces: the period map is shown to come from a family of Enriques surfaces, obtained as quotients of the Campedelli surface by an involution. The period map realises an isomorphism between a projective variety obtained by invariant theory, and the Baily-Borel compactification of an arithmetic quotient, in the same fashion as in the work of Matsumoto, Sasaki and Yoshida. The result is proved from scratch using traditional methods. As another consequence we determine properties of the mo…

Reduction and lifting of special metacyclic covers

Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective line with three branch points. Our results provide a classification of all special covers in terms of certain lifting data in characteristic p.

Automorphism groups of rational elliptic and quasi-elliptic surfaces in all characteristics

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.

Unprojection and deformations of tertiary Burniat surfaces

We construct a 4-dimensional family of surfaces of general type with p_g=0 and K^2=3 and fundamental group Z/2xQ_8, where Q_8 is the quaternion group. The family constructed contains the Burniat surfaces with K^2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on (\PP^1)^4 and we also give an action of Z/2xQ_8 on (\PP^1)^4 lifting the natural action on the surfaces. The strategy is the following. We consider an étale (Z/2)^3-cover T of a surface with p_g=0 and K^2=3 and assume that it may be embedded in a Fano 3-fold V. We construct…

K3 surfaces of finite height over finite fields

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for any such lifting, the endormorphism algebra of the transcendental cycles, as a Hodge module, is a CM field. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type.

CM liftings of K3 surfaces over finite fields and their applications to the Tate conjecture

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over finite fields. We prove every K3 surface of finite height over a finite field admits a characteristic 0 lifting whose generic fiber is a K3 surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a K3 surface over a finite field. To obtain these results, we construct an analogue of Kisin's algebraic group for a K3 surface of finite height, and construct characteristic 0 lif…

关于空间 $L_{m+1,n}$ 的算术与几何

On the arithmetic and geometry of spaces $L_{m+1,n}$

本文研究了特征 $p$ 下的射影线上对数微分形式的向量空间 $L_{m+1,n}$，并分类了特定情况下的这些空间。