Long Wang
We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic which is birational to an Enriques surface and whose automorphism group is discrete but not finitely generated.
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We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general $n$-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.
Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a one dimensional family of classical and supersingular Enriques surfaces with the automorphism group isomorphic to the symmetric group of degree five.
We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface.
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 determine the minimum positive entropy of complex Enriques surface automorphisms. This together with McMullen's work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of Enriques-Kodaira classification of complex surfaces.
It follows from an observation of A. Coble in 1919 that the automorphism group of an unnodal Enriques surface contains the $2$-congruence subgroup of the Weyl group of the $E_{10}$-lattice. In this article, we determine how much bigger the automorphism group of an unnodal Enriques surface can be. Furthermore, we show that the automorphism group is in fact equal to the $2$-congruence subgroup for generic Enriques surfaces in arbitrary characteristic (under the additional assumption that the Enriques surface is ordinary if the characteristic is $2$), improving the corresponding result of W. Bart…
We extend to arbitrary characteristic some known results about automorphisms of complex Enriques surfaces that act trivially on the cohomology or the cohomology modulo torsion.
We will show that there is a smooth complex projective surface, birational to some Enriques surface, such that the automorphism group is discrete but not finitely generated.
We classify the bi-canonical representations of finite automorphisms on Enriques surfaces. There are three types of non-trivial cases and examples are given explicitly by Horikawa models. In particular, finite non-semi-symplectic automorphisms exist only in orders 4 and 8. One corollary is that any finite cyclic subgroup in the automorphism group of an Enriques surface has order 1, 2, 3, 4, 5, 6, 8. Moreover, for two of the three types, a uniqueness theorem for maximal-dimensional families is given.
We present finite sets of generators of the full automorphism groups of three singular K3 surfaces, on which the alternating group of degree 6 acts symplectically. We also present a finite set of generators of the full automorphism group of an associated Enriques surface, on which the Mathieu group M_{10} acts.
Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially may be the minimal dynamical degree on Enriques surfaces and for whic…
An explicit Enriques surface with an automorphism of minimum entropy
本文借助计算机和椭圆纤维化方法，推导出复 Enriques 曲面上具有最小拓扑熵的 Oguiso-Yu 自同构的显式方程。
Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical Néron-Severi lattice of $Y$ induced by $f$. We apply a modification of McMullen's method to prove that the modulo-$2$ reduction $(p_f(x) \bmod 2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials $Φ_m$, where $m \leq 9$ and $m$ is odd. We study Enriques surfaces that realize modulo-$2$ reductions of $Φ_7$, $Φ_9$ and show that each of the five polynomials $(Φ_m(x) \bmod 2)$ is a factor of the modulo-$2$ reduction…
The Cone Conjecture for Enriques Surfaces in any Characteristic
论文证明了恩里克斯曲面的Morrison-Kawamata锥猜想，方法基于分析特定的双射映射。
Let $S$ be the (minimal) Enriques surface obtained from the symmetric quartic surface $(\sum_{i<j}x_ix_j)^2=kx_1x_2x_3x_4$ in $\mathbb{P}^3$ with $k\neq 0,4,36$, by taking quotient of the Cremona action $(x_i) \mapsto (1/x_i)$. The automorphism group of $S$ is a semi-direct product of a free product $\mathcal{F}$ of four involutions and the symmetric group $\mathfrak{S}_4$. Up to action of $\mathcal{F}$, there are exactly $29$ elliptic pencils on $S$.
We give conceptual proofs of some results on the automorphism group of an Enriques surface X, for which only computational proofs have been available. Namely, there is an obvious upper bound on the image of Aut(X) in the isometry group of X's numerical lattice, and we establish a lower bound for the image that is quite close to this upper bound. These results apply over any algebraically closed field, provided that X lacks nodal curves, or that all its nodal curves are (numerically) congruent to each other mod 2. In this generality these results were originally proven by Looijenga and Cossec-D…
On the cone conjecture for Enriques manifolds
本文证明当覆盖度为素数时，极一般Enriques流形满足Morrison-Kawamata锥猜想，并验证了与已知例子形变等价的极一般Enriques流形也满足该猜想。
Let Y be a complex Enriques surface whose universal cover X is birational to a general quartic Hessian surface. Using the result on the automorphism group of X due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of Y. A fundamental domain of the action of the automorphism group on the nef cone of Y is described explicitly. The list of elliptic fibrations on Y and the list of combinations of rational double points that can appear on a surface birational to Y are presented. As an application, a set of generators of the automorphism group of the generic Enriques s…
Enriques surfaces of zero entropy
本文对零熵Enriques曲面进行了分类，等价于对具有虚拟阿贝尔自同构群的Enriques曲面进行分类。
We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.
It is known that the dynamical degree, or equivalently, the topological entropy of an automorphism g of an algebraic surface S is lower semi-continuous when (S,g) varies in a algebraic family. In this paper we make a series of experiments confirming this behavior with the aim to realize small Salem numbers as the dynamical degrees of automorphisms of Enriques surfaces.
The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces
本文研究了Brandhorst和Shimada提出的$(τ,\overlineτ)$-generic Enriques曲面，给出了其非退化不变量的下界，证明多数情况下该不变量值为$10$，并首次发现具有无限自同构群且该不变量不为$10$的复Enriques曲面实例。
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 $…
We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Finally, we realize all types of Enriques surfaces with finite automorphism group over the prime fields $\mathbb{F}_p$ and $\mathbb{Q}$ whenever they exist.
Normal forms for quasi-elliptic Enriques surfaces and applications
本文通过建立拟椭圆Enriques曲面的正规形式，完成了由Kondo、Nikulin、Martin和Katsura-Kondo-Martin发起的具有有限自同构群的Enriques曲面的分类。
This is a brief introduction to the theory of Enriques surfaces over arbitrary algebraically closed fields. Some new results about automorphism groups of Enriques surfaces are also included.
Gap theorems and achirality for automorphisms of K3 surfaces and Enriques surfaces
本文研究K3和恩里克斯面自同构群的熵范数极值定理，并探讨其在 genus-1 纤维化下的非手性性质。
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…
We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds' method.
Orbits of smooth rational curves on Enriques surfaces
本文提出用 Nikulin 根不变量和 Vinberg 群计算 Enriques 曲面自同构群下光滑有理曲线轨道数的闭式公式。
Comparing Kahler cone and symplectic cone of one-point blowup of Enriques surface
本文研究Enriques曲面单点爆破的Kähler锥与辛锥的差异，证明其存在非Kähler辛形式，通过代数几何不变量定量比较两者区别，推进了$p_g=0$的Kähler曲面上辛通用复结构相关问题的研究。
The Enriques surface of minimal entropy
研究证明在特征2下存在唯一的恩里克斯表面具有动力学度λ₁₀，且在奇特征下无法实现，同时给出了该表面及相关自同构的显式方程。
We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality.
For an Enriques surface $S$, the non-degeneracy invariant $\mathrm{nd}(S)$ retains information on the elliptic fibrations of $S$ and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on $S$ together with a configuration of smooth rational curves, and gives a lower bound for $\mathrm{nd}(S)$. We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying $\mathrm{nd}(S)=10$ which are not general and with infini…
We point out an interesting relation between hypersurface elliptic singularities and log Enriques surfaces: with a few exceptions, every hypersurface elliptic singularity define some klt log Enriques surface $(S,Diff)$. In many cases, the log canonical cover of $(S,Diff)$ is a Du Val K3 surface with ``large'' nonsymplectic automorphism group.
Let $U\subset \mathbb P^N$ be a projective variety which is not a cone and whose hyperplane sections are smooth Enriques surfaces. We prove that the degree of a $U$ is at most 32 and the bound is sharp.
An action of a group $G$ on an Enriques surface $S$ is called Mathieu if it acts on $H^0(2K_S)$ trivially and every element of order 2, 4 has Lefschetz number 4. A finite group $G$ has a Mathieu action on some Enriques surface if and only if it is isomorphic to a subgroup of the symmetric group $\mathfrak{S}_6$ of degree 6 and the order $|G|$ is not divisible by $2^4$. Explicit Mathieu actions of the three groups $\mathfrak S_5, N_{72}$ and $\mathfrak A_6$, together with non-Mathieu one of $H_{192}$, on polarized Enriques surfaces of degree 30, 18, 10 and 6, respectively, are constructed witho…
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptiona…
We describe a period map for those simply connected Enriques surfaces in characteristic 2 whose canonical double cover is K3. The moduli stack for these surfaces has a Deligne-Mumford quotient that is an open substack of a $\mathbb P^1$-bundle over the period space. We also give some general results relating local and global moduli for algebraic varieties and describe the difference in their dimensions in terms of the failure of the automorphism group scheme to be reduced.
We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macrì's construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtai…
We show that there are exactly, up to isomorphisms, seven extremal log Enriques surfaces Z and construct all of them; among them types D_{19} and A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X_3 or X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface with Picard number 20 and discriminant 3 (resp. 4), which are called the most algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by Shioda-Inose and Vinberg).
In the present paper we describe the K3 surfaces admitting order 11 automorphisms and apply this to classify log Enriques surfaces of global index 11.
Throughout this paper, we work over ${\mathbb C}$, and $n$ is an integer such that $n\geq 2$. For an Enriques surface $E$, let $E^{[n]}$ be the Hilbert scheme of $n$ points of $E$. By Oguiso and Schröer, $E^{[n]}$ has a Calabi-Yau manifold $X$ as the universal covering space, $π:X\rightarrow E^{[n]}$ of degree $2$. The purpose of this paper is to investigate a relationship of the small deformation of $E^{[n]}$ and that of $X$ $({\rm Theorem}\ 1.1)$, the natural automorphism of $E^{[n]}$ $({\rm Theorem}\,1.2)$, and count the number of isomorphism classes of the Hilbert schemes of $n$ points of …
This paper is a survey about $K3$ surfaces with an automorphism and log rational surfaces, in particular, log del Pezzo surfaces and log Enriques surfaces. It is also a reproduction on my talk at "Mathematical structures of integrable systems and their applications" held at Research Institute for Mathematical Sciences in September 2018.
527 elliptic fibrations on Enriques surfaces
本文证明每个 Enriques 曲面在适当多重性计数下都恰好有 527 个椭圆纤维化同构类，并通过构造模空间研究遗忘映射的分歧行为，将其与 Morrison-Kawamata 锥猜想中的双曲体积联系起来。
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.
We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 o…
Nodal rational curves on Enriques surfaces of base change type
本文提出一种纯几何无格构造方法，证明了基变换型Enriques曲面上双截面通常是节点的，并展示了此类曲面承认形成秩8子群的节点有理曲线集。
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…
Given a smooth family F/Y of geometrically irreducible surfaces, we study sequences of arbitrarily near T-points of F/Y; they generalize the traditional sequences of infinitely near points of a single smooth surface. We distinguish a special sort of these new sequences, the strict sequences. To each strict sequence, we associate an ordered unweighted Enriques diagram. We prove that the various sequences with a fixed diagram form a functor, and we represent it by a smooth Y-scheme. We equip this Y-scheme with a free action of the automorphism group of the diagram. We equip the diagram with weig…
Bloch's conjecture for fake projective planes with an automorphism of order 3
本文证明了对某些一般型曲面上的对合在零维Chow群上作用为恒等，从而验证了Bloch猜想，并推广到某些伪射影平面。
We realize the logarithm of the third smallest known Salem number as the topological entropy of a K3 surface automorphism with a Siegel disk and a pointwisely fixed curve at the same time. We also show the logarithm of the Lehmer number, the smallest known Salem number, is not realizable as the topological entropy of any Enriques surface automorphism. These results are entirely inspired by McMullen's works and Mathematica programs.
Following a recent work of Oguiso, we calculate explicitly the groups of automorphisms and birational automorphisms on a Calabi-Yau manifold with Picard number two. When the group of birational automorphisms is infinite, we prove that the Cone conjecture of Morrison and Kawamata holds.
A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be either a torus, a K3 surface, an Enriques surface or a rational surface. Automorphisms of rational surfaces are quite mysterious and have been recently the object of intensive studies. In this paper, we construct several new examples of automorphisms of rational surfaces with positive topological entropy. We also explain how to define and to count parameters in families of birational maps of the complex projective plane and in families of rational surfaces.
The relative movable cone conjecture for K-trivial fibrations in varieties with well-clipped movable cones
本文证明了在纤维为商积的K-平凡纤维化中，弱相对Kawamata-Morrison可动锥猜想成立，且在特定条件下得出完整猜想，并推导出有限数量的极小模型。
We classify invariant probability measures for non-elementary groups of automorphisms, on any compact Kähler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also prove that except in certain rigid situations known as Kummer examples, there are only finitely many invariant, ergodic, probability measures with a Zariski dense support. If X is a K3 or Enriques surface, and the group does not preserve any algebraic subset, this leads to a complete description of orbit closures.
A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface. We deal with results obtained in this last case. After some recalls on birational maps, algebraic geometry and dynamic, we will speak about Bedford and Kim's works, but also about McMullen's work and more recently about Grivaux and the author's work. These notes have been written for four talks given at IMPA in february/march 2010.
We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperkähler manifolds and birational automorphism groups, as we shall see. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperkähler manifold of Picard number two. We will also discuss a similar conjectual relation together with exsistence of rational curve, expected by …
The big cone of every smooth projective surface $X$ admits the natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on $X$ to be numerically determined Weyl chambers. Such a criterion generalizes the results of Bauer-Funke on K3 surfaces to arbitrary smooth projective surfaces. In the last section, we study the relation between decompositions of the big cone and elliptic fibrations on Enriques surfaces.
Birational automorphism groups and the movable cone theorem for Calabi-Yau complete intersections of products of projective spaces
本文推广 Cantat–Oguiso 的 Coxeter 群方法，证明射影空间乘积中一般 Calabi-Yau 完全交的可移动锥猜想并刻画其双有理自同构群，还描述可移动锥边界及数值维数应用。
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