Numerical Campedelli surfaces with fundamental group of order 9

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.

KSBA空间上的Kappa类

Kappa classes on KSBA spaces

该研究在KSBA稳定簇及配对的模空间上定义了推广曲线模空间Miller-Morita-Mumford类的kappa类，并在含Burniat、Campedelli曲面的已知虚拟基本类存在情形中计算，还为Campedelli曲面求出了特定GIT商的Chow环。

Bloch's conjecture on surfaces of general type with an involution

In this short note we prove that the Bloch's conjecture holds for a surface of general type of numerical Godeaux type or some class of numerical Campedelli type, with geometric genus zero equipped with an involution, when the quotient of the surface by the involution is a rational surface.

Canonical maps of surfaces defined by Abelian covers

In this paper, we classified the surfaces whose canonical maps are abelian covers over $\mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $\mathbb{Z}_2^{\oplus 3}$ and give defining equations for Perssson's surface and Tan's surfaces with odd canonical degrees explicitly.

New surfaces with $K^2=7$ and $p_g=q\leq 2$

We construct smooth minimal complex surfaces of general type with $K^2=7$ and: $p_g=q=2,$ Albanese map of degree $2$ onto a $(1,2)$-polarized abelian surface; $p_g=q=1$ as a double cover of a quartic Kummer surface; $p_g=q=0$ as a double cover of a numerical Campedelli surface with $5$ nodes.

On Z/3-Godeaux surfaces

We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic fibrations with two multiplicity 3 fibres and one I_4 singular fibre. We study special such degenerations which have an involution, describing the corresponding Campedelli double plane construction. We also find some stable rational degenerations, some of which have more singularities, and one of which has a single 1/9(1,2) singularity, the minimal possible ind…

Numerical Godeaux surfaces with an involution

Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and K^2=1 and are usually called "numerical Godeaux surfaces". Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. with an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces, or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested…

具有大量互不相交(-2)-曲线的数值Godeaux曲面及其应用

Numerical Godeaux Surfaces with many disjoint $(-2)$-curves and Applications

本文证明数值Godeaux曲面的互不相交(-2)-曲线至多6条且界最优，将该结果应用于细化两类p_g=0的光滑极小一般型曲面对合的分类。

Mixed quasi-étale surfaces, new surfaces of general type with $p_g=0$ and their fundamental group

We call a projective surface $X$ mixed quasi-étale quotient if there exists a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C \rightarrow X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-étale surface. We study the mixed quasi-étale surfaces under the assumption that $(C\times C)/G^0$ has only nodes as singularities, where $G^0\triangleleft G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal reg…

A complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$

We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $π_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a $\mathbb{Q}$-Gorenstein smoothing theory.

希格斯丛与基本群概形

Higgs bundles and fundamental group schemes

本文利用数值有效性概念证明数值平坦希格斯向量丛构成Tannaka范畴，引入希格斯基本群概形，并将其性质与陈类消失猜想联系起来。

正特征曲线的驯服基本群概形

The tame fundamental group schemes of curves in positive characteristic

本文研究正特征代数闭域上光滑曲线的驯服基本群概形，发展余特化映射理论，证明其依赖曲线且可重构数值不变量。

Counting rational maps onto surfaces and fundamental groups

We consider the class of quasiprojective varieties admitting a dominant morphism onto a curve with negative Euler characteristic. The existence of such a morphism is a property of the fundamental group. We show that for a variety in this class the number of maps onto a hyperbolic curve or surfaces can be estimated in terms of the numerical invariants of the fundamental group. We use this estimates to find the number of biholomorphic automorphisms of complements to some arrangements of lines.

On the S-fundamental group scheme II

The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by V. Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial etale fundamental group.

Klein's Fundamental 2-Form of Second Kind for the $C_{ab}$ Curves

In this paper, we derive the exact formula of Klein's fundamental 2-form of second kind for the so-called $C_{ab}$ curves. The problem was initially solved by Klein in the 19th century for the hyper-elliptic curves, but little progress had been seen for its extension for more than 100 years. Recently, it has been addressed by several authors, and was solved for subclasses of the $C_{ab}$ curves whereas they found a way to find its individual solution numerically. The formula gives a standard cohomological basis for the curves, and has many applications in algebraic geometry, physics, and appli…

Gorenstein stable Godeaux surfaces

We classify Gorenstein stable numerical Godeaux surfaces with worse than canonical singularities and compute their fundamental groups.

Symplectic Lefschetz fibrations with arbitrary fundamental groups

In this paper we give an explicit construction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study the numerical properties of the sections in symplectic Lefschetz fibrations and their relation to the structure of the monodromy group.