The surface of Gauss double points

On the classification of surfaces of general type with non-birational bicanonical map and Du Val Double planes

We classify surfaces of general type whose bicanonical map is composed with a rational map of degree 2 onto a rational or ruled surface.

Explicit Resolutions of Double Point Singularities of Surfaces

Locally analytically, any isolated double point occurs as a double covering of a smooth surface. It can be desingularized via the canonical resolution, as it is well-known. In this paper we explicitly compute the fundamental cycle of both the canonical and minimal resolution of a double point singularity and we classify those for which the fundamental cycle differs from the fiber cycle. Finally we compute the conditions that a double point imposes to pluricanonical systems.

Smooth double covers of K3 surfaces

In this paper we classify the topological invariants of the possible branch loci of a smooth double cover $f:X\rightarrow Y$ of a K3 surface $Y$. We describe some geometric properties of $X$ which depend on the properties of the branch locus. We give explicit examples of surfaces $X$ with Kodaira dimension 1 and 2 obtained as double cover of K3 surfaces and we describe some of them as bidouble cover of rational surfaces. Then, we classify the K3 surfaces which admit smooth double covers $X$ satisfying certain conditions; under these conditions the surface $X$ is of general type, $h^{1,0}(X)=0$…

On singularity types of del Pezzo surfaces with rational double points in positive characteristic

In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo surface with rational double points whose anti-canonical members are all singular. As a corollary, we determine all singularity types of del Pezzo surfaces with rational double points which only appear in positive characteristic.

与Gushel-Mukai曲面相伴的双EPW六次超曲面

Double EPW sextics associated to Gushel-Mukai surfaces

本文刻画了相伴双EPW六次超曲面光滑的K3曲面，并由此在某些光滑双EPW六次超曲面上构造辛作用，同时给出Gushel-Mukai簇自同构群的界。

K3 double structures on Enriques surfaces and their smoothings

Let $Y$ be a smooth Enriques surface. A $K3$ carpet on $Y$ is a locally Cohen-Macaulay double structure on $Y$ with the same invariants as a smooth $K3$ surface (i.e., regular and with trivial canonical sheaf). The surface $Y$ possesses an étale $K3$ double cover $X \oversetπ \longrightarrow Y$. We prove that $π$ can be deformed to a family $\SX \longrightarrow \mathbf P^N_{T^*}$ of projective embeddings of $K3$ surfaces and that any projective $K3$ carpet on $Y$ arises from such a family as the flat limit of smooth, embedded $K3$ surfaces.

Moduli of double covers and degree one del Pezzo surfaces

Given a degree one del Pezzo surface with canonical singularities, the linear series generated by twice the anti-canonical divisor exhibits the surface as the double cover of the quadric cone branched along a sextic curve. It is natural to ask if this description extends to the boundary of a compactification of the moduli space of degree one del Pezzo surfaces. The goal of this paper is to show that this is indeed the case. In particular, we give an explicit classification of the boundary of the moduli space of anti-canonically polarized broken del Pezzo surfaces of degree one as double covers…

On the automorphisms of some $K3$ surface double cover of the plane

In this paper we study the automorphisms group of some $K3$ surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study the case of a $K3$ surface of Picard rank two such that there is a rational curve of degree $d$ which is tangent to the sextic in $d$ points.

Special systems through double points on an algebraic surface

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.

On nodal Enriques surfaces and quartic double solids

We consider the class of singular double coverings $X \to \PP^3$ ramified in the degeneration locus $D$ of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface $D$ one can associate an Enriques surface $S$ which is the factor of the blowup of $D$ by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent she…

Van Geemen--Sarti involutions and elliptic fibrations on K3 surfaces double cover of $\mathbb{P}^2$

In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen--Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen--Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.

Painlevé equations and deformations of rational surfaces with rational double points

In this paper, we show that the Bäcklund transformations of Painlevé equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlevé equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlevé equations.

Supersingular K3 surfaces in characteristic 2 as double covers of a projective plane

For every supersingular $K3$ surface $X$ in characteristic 2, there exists a homogeneous polynomial $G$ of degree 6 such that $X$ is birational to the purely inseparable double cover of a projective plane defined by $w^2=G$. We present an algorithm to calculate from $G$ a set of generators of the numerical Néron-Severi lattice of $X$. As an application, we investigate the stratification defined by the Artin invariant on a moduli space of supersingular $K3$ surfaces of degree 2 in characteristic 2.

On the projective normality of double coverings over a rational surface

We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $π$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+rπ^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.

del Pezzo曲面上存在哪些有理二重点？

Which rational double points occur on del Pezzo surfaces?

本研究确定了任意特征代数闭域上任意次数与皮卡秩的RDP del Pezzo曲面的所有有理二重点构型，推广了Du Val的经典结果，并给出了含非紧有理二重点的1次RDP del Pezzo曲面的简化方程。

The double of the doubles of Klein surfaces

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y\rightarrow X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if $X$ is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by $DX$. We describe how to use the double…

带有四个$D_4$型有理二重点的2次K3曲面的模空间

Moduli of K3 surfaces of degree 2 with four rational double points of type $D_4$

该研究确定了带四个$D_4$型有理二重点的2次格极化K3曲面模空间的Satake-Baily-Borel紧化为三维射影空间，并刻画了其自守形式分次环的生成元与关系。

On real anti-bicanonical curves with one double point on the $4$-th real Hirzebruch surface. II

A real 2-elementary K3 surfaces of type ((3,1,1),- id) yields a real anti-bicanonical curve s \cup A^\prime_1 (disjoint union) on the 4-th real Hirzebruch surface F_4 where s is the exceptional section of F_4 and the real curve A^\prime_1 has one real double point. We give a criterion (see Proposition 2.4) which determines whether the real double point is degenerate or not. One direction of the assertion of this proposition has already been proved in Lemma 4.6 of the preceding paper [9] (2015) by the author. In this paper we prove the inverse direction.

On Double Danielewski Surfaces and the Cancellation Problem

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.

K3 surfaces without section as double covers of Halphen surfaces, and F-theory compactifications

We construct several examples of genus-one fibered K3 surfaces without a global section with type $I_{n}$ fibers, by considering double covers of a special class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2. The resulting K3 surfaces have bisection geometries. F-theory compactifications on these K3 genus-one fibrations without a section times a K3 yield models that have $SU(n)$ gauge symmetries with a discrete $\mathbb{Z}_2$ symmetry.

The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial

We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory.

关于双重丹杰洛斯基表面的注记

A note on double Danielewski surfaces

本文修正了[arXiv:2403.02876]中定理3.11的证明，并提供了讨论各种情况的例子。

Double covers of Kummer surfaces

In this paper, we classify all the K3 surfaces covering a Kummer surface. Our classification is expressed in terms of period lattices and extends Morrison's criterion of K3 surfaces with a Shioda-Inose structure. Moreover, we list all the K3 surfaces covering a general Kummer surface and provide very geometrical examples of such a phenomenon.

$SU(n) \times \mathbb{Z}_2$ in F-theory on K3 surfaces without section as double covers of Halphen surfaces

We investigate F-theory models with a discrete $\mathbb{Z}_2$ gauge symmetry and $SU(n)$ gauge symmetries. We utilize a class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2, to yield genus-one fibered K3 surfaces with a bisection, but lacking a global section. We consider F-theory compactifications on these K3 surfaces times a K3 surface to build such models. We construct Halphen surfaces of index 2 with type $I_n$ fibers, and we take double covers of these surfaces to obtain K3 surfaces without a section with two type $I_n$ fibers, and K3 surfaces…

On Brauer groups of double covers of ruled surfaces

Let X be a smooth double cover of a geometrically ruled surface defined over a separably closed field of characteristic different from 2. The main result of this paper is a finite presentation of the 2-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Néron-Severi group of X. In particular, the result gives a central simple algebra representative for the unique nontrivial Brauer class on any Enriques surface. An example demonstrating the applications to the study of rational points is given.

双覆映射正规表面的去奇异化

Desingularization of double covers of regular surfaces

本文提出通过显式方程描述双覆映射正规表面的去奇异化算法。

Extra-special quotients of surface braid groups and double Kodaira fibrations with small signature

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…

Symplectic double for moduli spaces of G-local systems on surfaces

Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli space D(G,S) is the symplectic double of the Poisson moduli space of framed G-local systems on S. Its symplectic form is upgraded to a K2-symplectic structure for which the special coordinates are K2-Darboux coordinates.