Related papers: Surfaces on the Severi line in positive characteri…

We prove that for any prime number $p\ge 3$, there exists a positive number $\kappa_p$ such that $\chi(\mathcal{O}_X)\ge \kappa_pc_1^2$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$. In particular,…

Let $(S,L)$ be a polarized abelian surface of Picard rank one and let $\phi$ be the function which takes each ample line bundle $L'$ to the least integer $k$ such that $L'$ is $k$-very ample but not $(k+1)$-very ample. We use Bridgeland's…

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…

Let $(X,\Delta)$ be a pair. We study how the condition $\kappa(K_X + \Delta)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular…

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 $\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces…

We prove two new results for Seshadri constants on surfaces of general type. Let $X$ be a surface of general type. In the first part, inspired by \cite{B-S}, we list the possible values for the multi-point Seshadri constant…

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…

Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface $Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves contained in $X$…

Consider a closed connected hypersurface in $\mathbb{R}^n$ with constant signature (k,l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides $\mathbb{R}^n$ into two pieces. We prove that one…

Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot \pi^*L_S - \alpha \cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $\pi:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with…

Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if…

Let $A$ be an abelian scheme of dimension at least four over a $\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$…

This thesis is devoted to the study of abelian automorphism groups of surfaces and $3$-folds of general type over complex number field $\Bbb C$. We obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of general type…

In this paper we continue the study of algebraic fundamentale group of minimal surfaces of general type S satisfying K_S^2<3\chi(S). We show that, if K_S^2= 3\chi(S)-1 and the algebraic fundamental group of S has order 8, then S is a…

In this work we present new results to produce an algorithm that returns, for any fixed pair of natural integers $K^2$ and $\chi$, all regular surfaces $S$ of general type with self-intersection $K_S^2=K^2$ and Euler characteristic…

We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and…

Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $\varphi \colon X \to Y$, we give an upper bound on the minimum of the degrees…

In this note, we construct a minimal surface of general type with geometric genus p g = 4, self-intersection of the canonical divisor K^2 = 32 and irregularity q = 1 such that its canonical map is an abelian cover of degree 16 of P^1 x P^1.

A conjecture of Colliot-Th\'{e}l\`{e}ne predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a…

Let $S$ be an abelian surface over an algebraically closed field $k$ with characteristic different from $2$ and $3$, and $\mathcal{L}$ a symmetric ample line bundle defining a polarisation of type $(1,3)$. Then the linear system…