F. Bogomolov, Yu. Tschinkel
Let $X$ be an Enriques surface defined over a number field $K$. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.
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We prove the rationality of the coarse moduli spaces of Coble surfaces and of nodal Enriques surfaces over the field of complex numbers.
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.
It was Fano who first classified Enriques-Fano threefolds. However his arguments appear to contain several gaps. In this paper, we will verify some of his assertions through the use of modern techniques.
Given d in IN, we prove that any polarized Enriques surface (over any field of characteristic different from 2 or with a smooth K3 cover) of degree greater than 12d^2 contains at most 12 rational curves of degree at most d. For d>2 we construct examples of Enriques surfaces of high degree that contain exactly 12 rational degree d curves.
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。
Singular rational curves on Enriques surfaces
本文证明非常一般的 Enriques 曲面在所有满足模 4 同余于 1 的正整数亏格上，均存在指定 $ϕ$-不变量的奇异有理曲线。
Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces
本文证明了基变换型Enriques曲面上任意亏格曲线的Severi簇可数族非空，并构造了有理椭圆曲面上特殊超丰对数Severi簇的例子。
Rational curves and Seshadri constants on Enriques surfaces
本文证明了在非常一般的 Enriques 曲面上有理曲线的类总是 $2$-可除的，并据此得出其上任何 big 和 nef 线丛的 Seshadri 常数与 Cossec 引入的 $ϕ$-函数值相等的结论。
We study the arithmetic of Enriques surfaces whose universal covers are singular K3 surfaces. If a singular K3 surface X has discriminant d, then it has a model over the ring class field d. Our main theorem is that the same holds true for any Enriques quotient of X. It is based on a study on Neron-Severi groups of singular K3 surfaces. We also comment on Galois actions on divisors of Enriques surfaces.
This paper develops families of complex Enriques surfaces whose Brauer groups pull back identically to zero on the covering K3 surfaces. Our methods rely on isogenies with Kummer surfaces of product type. We offer both lattice theoretic and geometric constructions. We also sketch how the construction connects to string theory and Picard-Fuchs equations in the context of Enriques Calabi-Yau threefolds.
Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two
本文在特征二中分类了具有九个光滑有理曲线构型的奇异恩里克斯曲面，通过收缩曲线得到 Q_l-上同调同构于射影平面的 ADE 奇点曲面，并推广至经典恩里克斯曲面及积分模型。
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.
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 construct an Enriques surface X over Q with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.
Idoneal genera and K3 surfaces covering an Enriques surface
本文推广 Euler idoneal 数定义 idoneal 亏格，并用 Smith–Minkowski–Siegel 质量公式完全枚举，进而分类覆盖 Enriques 曲面的 K3 曲面的超越格。
Nodal Enriques surfaces are Reye congruences
本文证明了任何包含光滑有理曲线的经典恩里克斯曲面均可实现为雷耶线汇，揭示了这两类代数几何对象之间的对应关系。
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 define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana and Corvaja--Zannier holds for Enriques surfaces, as well as K3 surfaces of Picard rank greater than 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.
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.
Extendability of group actions on K3 or Enriques surfaces
本文研究了K3和恩里克斯表面群作用的可扩展性，提出了基于ℓ-adic上同调群作用的准则，并推广了Galois作用的可扩展性结果。
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。
Log Enriques surface is a generalization of K3 and Enriques surface. We will classify all the rational log Enriques surfaces of rank 18 by giving concrete models for the realizable types of these surfaces.
In 2011, Vàrilly-Alvarado and the last author constructed an Enriques surface $X$ over $\mathbb{Q}$ with an étale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the nontrivial Brauer class of $X_{\bar{\mathbb{Q}}}$ does not descend to $\mathbb{Q}$. Together with the results of Vàrilly-Alvarado and the last author, this proves that the Brauer-Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces. The methods of this paper build on the ideas in several recent papers by the last autho…
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 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.
Log Enriques surfaces with delta=1 are classified.
Enriques surfaces of non-degeneracy 3
本文分类了Enriques曲面上不可延拓的半纤维3-序列，并证明在特征不为2时每个Enriques曲面都容许4-序列，从而给出其与Enriques六次曲线和Castelnuovo五次曲面的双有理联系。
We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.
Mondromy and deformation-and-specialization arguments are used to produce a pencil of Enriques surfaces with no section and index one. The same technique completes the proof that the witness curves in "Rational connectivity ..." depend on the relative dimension.
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.
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.
Log Enriques surfaces with delta=2 are classified.
We give a different proof of Nuer's result on the exsistence of stable sheaves on Enriques surfaces.
We compute the monodromy groups of real Enriques surfaces of hyperbolic type. The principal tools are the deformation classification of such surfaces and a modified version of Donaldson's trick, relating real Enriques surfaces and real rational surfaces.
We determine the necessary and sufficient conditions on the entries of the intersection matrix of the transcendental lattice of a K3 surface for the K3 surface to doubly cover an Enriques surface.
We shall study a necessary and sufficient condition for the existence of stable sheaves on arbitrary Enriques surfaces.
We show that the (twisted) derived category "recognizes" the three different kinds of Enriques surfaces in characteristic 2
We study Ulrich bundles and their moduli on unnodal Enriques surfaces. In particular, we prove that unnodal Enriques surfaces are of wild representation type by constructing moduli spaces of stable Ulrich bundles of arbitrary rank and arbitrarily large dimension.
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.
In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that…
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…
This paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that Aut(X) is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.
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.
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 show that there is only one log Enriques surface of index 7 and type $A_{15}$.
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…
On the Fano dimension of an Enriques surface
本文通过构造Fano四维流形，将一般Enriques曲面的Fano维度从六降低到四。
The three pencils of K3 surfaces of minimal discriminant whose general element covers at least one Enriques surface are Kondō's pencils I and II, and the Apéry--Fermi pencil. We enumerate and investigate all Enriques surfaces covered by their general elements.
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.
We complete our study of linear series on curves lying on an Enriques surface by showing that, with the exception of smooth plane quintics, there are no exceptional curves on Enriques surfaces, that is, curves for which the Clifford index is not computed by a pencil.
On extra-special Enriques surfaces
本文细化Cossec和Dolgachev对超特殊恩里克斯曲面的分类，并给出完整而简洁的新证明。
In this paper we discuss the number of Enriques quotients of a fixed K3 surface. We prove the finiteness and unboundedness of the number. We also show an example of Kummer surface of product type where we can successfully classify all the Enriques quotients.
We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero. The key observation is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in IP^5, even in characteristic 2.
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.
We present the classification of involutions on Enriques surfaces. We classify those into 18 types with the help of the lattice theory due to Nikulin. We also give all examples of the classification.
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 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.
Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our study of Brill-Noether theory of curves on Enriques surfaces (reference [KL1]) and of Enriques-Fano threefolds (reference [KLM]).
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.
Convex Fujita numbers and the Kodaira-Enriques classification of surfaces
本文探讨极小光滑射影曲面在Kodaira-Enriques分类中凸Fujita数的取值范围。
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