Related papers: Surfaces on the Severi line

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational…

We classify the minimal surfaces of general type with $K^2 \leq 4\chi-8$ whose canonical map is composed with a pencil, up to a finite number of families. More precisely we prove that there is exactly one irreducible family for each value…

We give a list of possibilities for surfaces of general type with $p_g=0$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$ and $S/i$ is not rational. Some examples with $K^2=4,...,7$ are constructed as…

Let $S$ be a surface isogenous to a product of curves of unmixed type. After presenting several results useful to study the cohomology of $S$ we prove a structure theorem for the cohomology of regular surfaces isogenous to a product of…

Let (A,L) be a principally polarized abelian surface of type (1,3). The linear system |L| defines a 6:1 covering of A onto P2, branched along a curve B of degree 18 in P2. The main result of the paper is that for general (A,L) the curve B…

Let $(S,L)$ be a general primitively polarized $K3$ surface of genus $g$. For every $0\leq \delta \leq g$ we consider the Severi variety parametrizing integral curves in $|L|$ with exactly $\delta$ nodes as singularities. We prove that its…

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic…

We prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic. It can be viewed as a relative version of the classical Noether inequality for…

In this paper we study the cohomology of smooth projective complex surfaces $S$ of general type with invariants $p_g = q = 2$ and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the…

We carry out an analysis of the canonical system of a minimal complex surface of general type with irregularity q>0. Using this analysis we are able to sharpen in the case q>0 the well known Castelnuovo inequality K^2>=3p_g+q-7. Then we…

Let S be a minimal surface of general type with $p_g(S)=0$ and such that the bicanonical map $\phi:S\to \pp^{K^2_S}$ is a morphism: then the degree of $\phi$ is at most 4 and if it is equal to 4 then $K^2_S\le 6$. Here we prove that if…

We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C x C)/G of the "square" of a curve by a finite group G of automorphisms that contains elements not preserving the factors. We study them through the…

A complex surface $S$ is said to be isogenous to a product if $S$ is a quotient $S=(C_1 \times C_2)/G$ where the $C_i$'s are curves of genus at least two, and $G$ is a finite group acting freely on $C_1 \times C_2$. In this paper we…

We study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by C. Rito. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus…

Let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and $K^{2}_{S}=4$. Assume the bicanonical map $\varphi$ of $S$ is a morphism of degree $4$ such that the image of $\varphi$ is smooth. Then we prove that the surface $S$ is a…

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus $g$.We prove a quadratic upper bound on the genus $g$, i.e., $g\leq h\big(\chi(\mathcal{O}_S)\big)$, where…

We give two explicit versions of the decomposition theorem of Beilinson, Bernstein and Deligne applied to the universal family of quartic surfaces of $\mathbb{P}^3$. The starting point of our investigation is the remark that the nodes of a…

Let $V$ be a smooth quasi-projective complex surface with compactification $(X,D)$ and set $\overline P_1(V):=h^0(X,K_X+D)$, $\overline q(V):=h^0(X,\Omega^1_X(\log D))$. We prove that $\overline P_1(V)\ge \overline q(V)-1$ if $V$ has…

In this paper, we study the Gieseker moduli space $\mathcal{M}_{1,1}^{4,3}$ of minimal surfaces with $p_g=q=1, K^2=4$ and genus 3 Albanese fibration. Under the assumption that direct image of the canonical sheaf under the Albanese map is…

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with…