On the number of Enriques quotients of a K3 surface
Hisanori Ohashi
本文讨论了固定K3曲面的Enriques行列式的个数。我们证明了这个数的有限性和无界性。我们还展示了一个例子，库默尔表面的产品类型，我们可以成功地分类所有的恩里克斯concurents。
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.
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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…
Quotients of Calabi-Yau Generalized Fermat manifolds and Log Enriques varieties
本文通过刻画Calabi-Yau型广义费马流形的无不动点自同构商，构造了对数Enriques簇与Enriques流形，证明其为已知两类Calabi-Yau型Enriques流形族中一类的子族，且与另一类的第二Betti数不同。
The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces
本研究引入正特征下代数群概形的极大幂幺有限商，发现其应用于皮卡概形可编码异常挠率，借助例外恩里克斯曲面理论构造出存在该异常挠率的整法诺三维簇，打破了德尔佩佐曲面挠率为零的常规认知。
Low codimension Fano--Enriques threefolds
本文研究了具有扭除子的Fano三重（Fano-Enriques）.由于这个扭除子，它们可以被描述为有限阿贝尔群作用的Fano三重项。我们从里德、弗莱彻和阿尔特诺克的法诺三重表开始，检查他们中的哪一个承认这样的行动与法诺-恩里克斯商。
We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In 'generic' cases this gives a bijection between the set Enr(X) of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank 20 we prove that the fibres of the map from Enr(X) to Br(X)[2] above the non-zero points have the same order.
We give an explicit description of the Godeaux surfaces that admit an involution such that the quotient surface is birational to an Enriques surface; these surfaces give a 6-dimensional unirational irreducible subset of the moduli space of surfaces of general type. In addition, we describe the Enriques surfaces that are birational to the quotient of a Godeaux surface by an involution and we show that they give a 5-dimensional unirational irreducible subset of the moduli space of Enriques surfaces. Finally, by degenerating our description we obtain some examples of non-normal stable Godeaux sur…
The quotient space of a $K3$ surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, we will study finite groups which act faithfully on $K3$ surfaces such that the quotient space are smooth. In particular, we will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a $K3$ surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.…
Descent of tautological sheaves from Hilbert schemes to Enriques manifolds
本文研究 K3 曲面双重覆盖 Enriques 曲面时，奇数点 Hilbert 概形的商为 Enriques 流形，并构造其上的斜率稳定层，探讨其性质。
On the Brauer group of Enriques surfaces
设S是一个复Enriques曲面;它是一个K3曲面X与一个无不动点对合的商。Brauer群Br（S）有唯一的非零元素。我们在Br（X）中描述了它的拉回，并证明了它是平凡的曲面S在Enriques曲面的模空间中构成一个可数的超曲面并.
In this short note we prove that an involution on certain examples of surfaces of general type with $p_g=0$, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is an Enriques surface and show that the Bloch's conjecture holds for such surfaces.
We compute the intersection Betti numbers of the GIT model of the moduli space of numerically polarized Enriques surfaces of degree 2. The strategy of the cohomological calculation relies on a general method developed by Kirwan to compute the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities, called Kirwan blow-up.
Enriques surfaces with eight nodes
节点Enriques曲面最多可以有8个节点。我们给出了一个明确的描述Enriques曲面与8个节点，表明它们是产品的椭圆曲线的一个组同构于$\Z_2^2$或$\Z_2^3$自由作用在余维1的直子。我们使用该结果来表明，如果$S$是具有$p_g=0$的一般类型的极小曲面，使得双典范映射的像对于Enriques曲面是双有理的，则$K^2_S=3$和双典范映射是2次态射。
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.
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。
The classification of log Enriques surfaces of rank 18
Log Enriques曲面是K3曲面和Enriques曲面的推广。我们将分类所有的合理的日志恩里克表面的秩18通过给出具体的模型，这些表面的可实现的类型。
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.
Classification of log Enriques surfaces with delta=1
Delta=1的Log Enriques曲面被分类。
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.
Moduli and periods of simply connected Enriques surfaces
本文描述了特征为2的单连通Enriques曲面的周期映射，其典型双覆盖为K3。这些曲面的模栈有一个Deligne-Mumford商，它是周期空间上$\mathbb P^1$ -丛的开子栈。我们还给出了一些一般性的结果有关的地方和全球模的代数簇和描述的差异，在他们的尺寸方面的失败的自同构群计划减少。
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.
Classification of log Enriques surfaces with delta=2
对Delta=2的Log Enriques曲面进行了分类。
The rationality of the moduli spaces of Coble surfaces and of nodal Enriques surfaces
证明了复数域上Coble曲面和节点Enriques曲面的粗模空间的合理性。
We give a different proof of Nuer's result on the exsistence of stable sheaves on Enriques surfaces.
Which Singular K3 Surfaces Cover an Enriques Surface
给出了K3曲面的超越格的交矩阵的元素使K3曲面双重覆盖Enriques曲面的充要条件.
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…
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。
Enriques surfaces covered by Jacobian Kummer surfaces
本文对K3-覆盖为固定Picard-广义Jacobian Kummer曲面的Enriques曲面进行了分类。有31个这样的表面。我们描述的自由对合，使这些恩里克表面明确。作为一个双积，我们证明了Aut（X）是由2阶元生成的，这是S.近藤
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 $\bar{Y}$.
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.
Enriques Surfaces and jacobian elliptic K3 surfaces
本文提出了一种新的Enriques曲面的尺规作图法。它的出发点是K3表面的雅可比椭圆纤维化所产生的有理椭圆曲面的二次基变化。以这种方式获得的Enriques表面的特征在于椭圆纤维化的有理曲线作为平分分裂成两个部分的覆盖K3表面上。该构造可用于研究具有特定自同构的Enriques曲面。它还使我们能够回答一个问题的博维尔恩里克表面的布劳尔集团表现出一种特殊的行为。在即将发表的论文中，我们将研究我们的构造的算术后果。
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 $A_{15}$.
Monodromy groups of real Enriques surfaces
计算了实双曲型Enriques曲面的单值群。主要工具是变形分类等表面和修改版本的唐纳森的技巧，有关真正的恩里克斯表面和真正的理性表面。
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.
Brill-Noether theory of curves on Enriques surfaces, II. The Clifford index
我们完成了我们的研究线性系列的曲线躺在恩里克斯曲面表明，除了光滑的平面五次曲线，有没有例外的曲线恩里克斯曲面，也就是说，曲线的克利福德指数不是由铅笔计算。
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。
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.
Arithmetic Moduli and Lifting of Enriques Surfaces
构造了正特征及整数上的Enriques曲面的模空间，并确定了它的局部结构和整体结构.作为一个应用，我们展示了提升的Enriques表面的特征零。关键的观察是，恩里克曲面的正则双覆盖对于IP^5中的三个二次曲面的完全交是双有理的，即使特征为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.
Classification of involutions on Enriques surfaces
给出了Enriques曲面上对合的分类。我们将其分为18种类型的晶格理论的帮助下，由于Nikulin。我们还给出了所有分类的例子。
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.
Numerical trivial automorphisms of Enriques surfaces in arbitrary characteristic
我们推广到任意特征的一些已知结果的自同构复杂的Enriques曲面，平凡的上同调或上同调模挠。
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.
A sharp vanishing theorem for line bundles on K3 or Enriques surfaces
设$L$是K3或Enriques曲面上的线丛。我们给$H^1(L)$的消灭定理，不像大多数消灭定理，给出了必要和充分的几何条件消失。这个结果在我们研究Enriques曲面上曲线的Brill-Noether理论（参考文献[KL 1]）和Enriques-Fano三重理论（参考文献[KLM]）中是必不可少的。
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。
In this note, we use crystalline methods and the Tate-conjecture to give a short proof that the Picard rank of an Enriques surface is equal to its second Betti number.
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.
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。
Grothendieck-Riemann-Roch and the moduli of Enriques surfaces
我们给出了Borcherds定理Enriques曲面的模空间是拟仿射的一个简短而经典的证明。在我们的证明中，Borcherds乘积的使用被Grothendieck-Riemann-Roch定理的应用所取代。
We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond.
Elliptic Gorenstein singularities, log canonical thresholds and log Enriques surfaces
我们指出了超曲面椭圆奇点和log Enriques曲面之间的一个有趣的关系：除了少数例外，每个超曲面椭圆奇点定义了一些klt log Enriques曲面$(S,Diff)$。在许多情况下，$(S,Diff)$的对数典范覆盖是一个具有“大”非辛自同构群的Du Val K3曲面。
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