Numerical Godeaux surfaces with an involution

An 8-dimensional family of simply connected Godeaux surfaces

In this paper we describe a construction method for numerical Godeaux surfaces based on homological algebra. We show the existence of an 8-dimensional locally complete family of simply connected numerical Godeaux surfaces.

Automorphisms of numerical Godeaux surfaces with torsion of order 3, 4, or 5

We compute the automorphisms groups of all numerical Godeaux surfaces, i.e. minimal smooth surfaces of general type with K^2 = 1 and p_g = 0, with torsion of the Picard group of order νequals 3, 4, or 5. We present explicit stratifications of the moduli spaces whose strata correspond to different automorphisms groups. Using the automorphisms computation, for each value of νwe define a quotient stack, and prove that for ν= 5 this is indeed the moduli stack of numerical Godeaux surfaces with torsion of order 5. Finally, we describe the inertia stacks of the three quotient stacks.

$\mathbb Z/2$-Godeaux 表面

$\mathbb Z/2$-Godeaux surfaces

研究$\mathbb Z/2$-Godeaux表面的模空间，证明其为不可约且无理的8维空间，并展示其拓扑基本群为$\mathbb Z/2$。

Marked Godeaux surfaces with special bicanonical fibers

In this paper we study marked numerical Godeaux surfaces with special bicanonical fibers. Based on our construction method of marked Godeaux surfaces we give a complete characterization for the existence of hyperelliptic bicanonical fibers and torsion fibers. Moreover, we describe how the families of Reid and Miyaoka with torsion $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/5\mathbb{Z}$ arise in our homological setting.

Bloch's conjecture for surfaces with involutions and of geometric genus zero

Let $S$ be a smooth projective surface with $p_g=0$, let $ι$ be a regular involution acting on $S$, and let $W$ be the resolution of singularities of the quotient surface $S/ι$. In the paper we prove that Bloch's conjecture holds for the surface $S$ if and only if it holds for the surface $W$. This yields Bloch's conjecture for all surfaces $S$ whenever the same conjecture is true for the desingularized quotient $W$. In particular, Bloch's conjecture holds true for all numerical Godeaux surfaces with involutions, a "half" of Campedelli surfaces with involutions, the surface of Craighero and Ga…

A simply connected numerical Godeaux surface with ample canonical class

We prove that a recent construction of a numerical Godeaux surface due to P. Craighero and R. Gattazzo is simply connected, and show how to realize their construction as a double plane. By proving that the surface contains no (-2)-curves, we obtain that this is the first example of a simply connected surface with vanishing geometric genus and ample canonical class.

奇异戈代克斯曲面的计算与一个新的显式伪二次曲面（附克里斯蒂安·格莱斯纳和诺亚·鲁兰德的附录）

Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

提出检测代数簇族高度奇异成员的计算方法，用于戈代克斯曲面获显例，构造出特定奇异戈代克斯曲面，其\(\mathbb{Z}/2 \times \mathbb{Z}/4\)阿贝尔覆盖是伪二次曲面，给出首个非曲线乘积商的伪二次曲面显式构造。

On the blowups of numerical Godeaux surfaces

We give a short proof of the following result: Let $X$ be a complex surface of general type. If the canonical divisor of the minimal model of $X$ has selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface. Our proof is the natural extension of the argument given in our paper in Inventiones Mathematicae (1989) for the case when $X$ is minimal. This argument also gives information about the non--existence of certain smooth embeddings of $2$--spheres in $X$, if $X$ has geometric genus zero.

Pluricanonical systems on surfaces with small K^2

We prove that the bicanonical system on a surface of general type with K^2=4 has no base components and describe clusters contracted by 4K_X for a numerical Godeaux surface and 3K_X for a numerical Campedelli surface.

Bloch's conjecture for the surface of Craighero and Gattazzo

The goal of this paper is to prove Bloch's conjecture for the numerical Godeaux surface constructed by P. Craighero and R. Gattazzo.

Involutions on numerical Campedelli surfaces

Numerical Campedelli surfaces are minimal surfaces of general type with p_g=0 (and so q=0) and K^2=2. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Campedelli surfaces with an involution, i.e. an automorphism of order 2. First we show that an involution on a numerical Campedelli surface S has either four or six isolated fixed points, and the bicanonical map of S is composed with the involution if and only if the involution has six isolated fixed points. Then we study in detail each of the possible cases, desc…

Involutions on surfaces with $p_g=q=1$

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i.$ A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g=q=1$ and $K^2=7$ having birational bicanonical map is obtained.

A simply connected numerical Campedelli surface with an involution

We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=1$. In order to construct the example, we combine a double covering and $\mathbb{Q}$-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surf…

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

We study automorphisms of the Hilbert scheme of $n$ points on a generic projective K3 surface $S$, for any $n \geq 2$. We show that the automorphism group of $S^{[n]}$ is either trivial or generated by a non-symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any $n \geq 2$, there exist infinite values for the degree of $S$ such that $S^{[n]}$ admits a non-natural involution. This provides a generalization of results by Boissière--Cattaneo--Nieper-Wisskirchen--Sarti…

Theta特征与某些Kummer型簇上[-1]的不动点轨迹

Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type

该文研究Kummer型超凯勒簇上辛对合不动点轨迹的组合性质，证明带(1,d)极化阿贝尔曲面对应的两类Kummer型超凯勒簇同构，分析d为奇时[-1_A]孤立不动点的双射与theta特征的关联，并给出相关计数公式的数值表。

On the numerical evaluation of algebro-geometric solutions to integrable equations

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions an…

Modularity of some non-rigid double octic Calabi-Yau threefolds

In this paper we discuss four methods of proving modularity of Calabi--Yau threefolds with $h^{12}=1$: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livné-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove modularity of 17 out of 18 double octic Calabi-Yau threefolds for which "numerical evidence of modularity" was found in the second Author's thesis.…