On extra-special Enriques surfaces
Gebhard Martin, Giacomo Mezzedimi, Davide Cesare Veniani
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。
我们细化了Cossec和Dolgachev对超特殊恩里克斯曲面的分类，给出了一个完整且简洁的证明。
We refine Cossec and Dolgachev's classification of extra-special Enriques surfaces, providing a complete and concise proof.
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We give an explicit construction and clasification of some very special sort of Enriques surfaces in characteristic two. This proves the existence of some of the surfaces that were called ``extra-special'' by Cossec and Dolgachev in their book on the subject, and whose existence was left open by them. These surfaces coincide with a class of surfaces that are introduced in recent work by Ekedahl and Shepherd--Barron.
New examples of geometrically special varieties: K3 surfaces, Enriques surfaces, and algebraic groups
本文验证椭圆K3曲面和代数群是Javanpeykar-Rousseau意义下的几何特殊簇，并证明附加假设下移除余维≥2闭子集后该特殊性仍保持，提供了几何特殊簇的新例子。
We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X
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.
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.
Bridgeland-Enriques general K3 surfaces
受 K3 曲面上 Enriques 范畴和 Bridgeland 稳定性条件不变量的启发，引入 Bridgeland-Enriques 一般 K3 曲面的概念，并探讨其与特殊 Gushel-Mukai 三维簇和 Hodge 特殊 Gushel-Mukai 四维簇的联系。
Two indecomposable Enriques categories
本研究证明四次双立体或特殊Gushel-Mukai三维簇的Kuznetsov分量无非平凡容许子范畴，涉及Enriques范畴的不可分解性问题。
After an Introduction to the themes of Enriques surfaces and Rationality questions, the Artin-Mumford counterexample to Lueroth problem is revisited. A construction of it is given, which is related in an explicit way to the geometry of Enriques surfaces, more precisely to the special family of Reye congruences and their classical geometry.
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。
Enriques surfaces with non-generic non-degeneracy
本文研究了非典型非退化恩里科斯表面的非退化不变量，发现部分表面族具有nd(Y)=9。
In this article we explicitly compute equations of an Enriques surface via the involution on a K3 surface. We also discuss its tropicalization and compute the tropical homology, thus recovering a special case of the result of \cite{IKMZ}, and establish a connection between the dimension of the tropical homology groups and the Hodge numbers of the corresponding algebraic Enriques surface.
On two families of Enriques categories over K3 surfaces
本文研究K3表面上的两个Enriques范畴家族的模空间，涉及四次双曲面和特殊Gushel-Mukai三维流形的构造，恢复经典几何构造，并描述模空间的奇异位点和O'Grady超Kähler十维空间的birational自映射。
The Weakly Special Conjecture contradicts orbifold Mordell, and hence the abc conjecture
该研究通过构造特定的弱特殊三维流形，证明了弱特殊猜想与orbifold莫尔德猜想矛盾，进而影响abc猜想，并展示了Enriques和K3表面的非可除退化情况。
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…
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.
Let S be a smooth algebraic surface satisfying the following property: H^i(\oc_S(C))=0 (i=1,2) for any irreducible and reduced curve C of S. The aim of this paper is to provide a characterization of special linear systems on S which are singular along a set of double points in general position. As an application, the dimension of such systems is evaluated in case S is an Abelian, an Enriques, a K3 or an anticanonical rational surface.
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…
We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.
Algebraic surfaces of general type with $q=0$, $p_g=2$ and $K^2=1$ were described by Enriques and then studied in more detail by Horikawa. In this paper we consider a $16$-dimensional family of special Horikawa surfaces which are certain bidouble covers of $\mathbb{P}^2$. The construction is motivated by that of special Kunev surfaces which are counterexamples for infinitesimal Torelli and generic global Torelli problem. The main result of the paper is a generic global Torelli theorem for special Horikawa surfaces. To prove the theorem, we relate the periods of special Horikawa surfaces to the…
Let $S$ be a surface with $p_g(S)=q(S)=0$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that $S$ supports special (often stable) Ulrich bundles of rank $2$, extending a recent result by A. Beauville. Moreover, we show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$ except for very few cases. We also show that the same is true for linearly normal non-special surface in $\mathbb P^4$ of degree at least $4$, Enriques surface and anticanonical r…
Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the Universal Severi-Enriques variety $\mathcal V$ on $\mathcal S\to\mathbb A^1$. The general fibre of $\mathcal V$ is the variety of curves on $\mathcal S_t$ in the linear system $|\mathcal O_{\mathcal S_t}(n)|$ with $k$ cusps and $δ$ nodes as singularities. Our problem is to find all irreducible components of the special fibre of $\mathcal V$. In this paper, we…
We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$, where $Σ_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if a finite group $G$ admits a diagonal double Kodaira structure, then $|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two $3$-dimen…
Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients
研究亏格为\(g\)的紧致黎曼曲面上双股全辫群的有限循环、二面体和超特殊商群数量，在商映射特定条件下计算。并将代数结果用于构造一般型光滑曲面，得到两个三维族的极小曲面，其有相同双正则不变量和贝蒂数，但第一同调群挠部分不同。
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