相关论文: Singularities of divisors of low degree on abelian…

It is known by results of Koll\'ar, Ein, Lazarsfeld, Hacon and Debarre that divisors representing principal and other low degree polarizations on abelian varieties have mild singularities. In this note we extend such results to…

Pareschi showed that an ample divisor of degree $d$ on a simple abelian variety of dimension $g$ has mild singularities when $d<g$. We extend his result to general polarized abelian varieties. We also show that ample divisors of degree $3$…

We provide numerical conditions for a polarized abelian threefold $(A,L)$ to have simple syzygies, in terms of property $(N_p)$ and the vanishing of Koszul cohomology groups $K_{p,1}$. We rely on a reduction method of…

In this paper we consider ideal sheaves associated to the singular loci of a divisor in a linear system $|L|$ of an ample line bundle on a complex abelian variety. We prove an effective result on their (continuous) global generation, after…

In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of…

We study the special fibers of a certain class of absolutely simple abelian varieties over number fields with endomorphism rings $\bz$ and possessing $l$-adic monodromy groups of the least possible rank. We also study the Dirichlet density…

We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is…

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to…

We characterize simple complex abelian varieties and simple abelian surfaces in terms of primitivity of translation automorphisms. Applying this together with a result due to Diller and Favre, we then classify all primitive birational…

The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F-polarization on every abelian variety in the isogeny class.

We refine and generalize the results of K. E. Lauter and E. W. Howe on principal polarizations on products of abelian varieties over finite fields. Firstly, we study the reasons for the absence of an irreducible principal polarization in…

The degree of a curve $C$ in a polarized abelian variety $(X,\lambda)$ is the integer $d=C\cdot\lambda$. When $C$ generates $X$, we find a lower bound on $d$ which depends on $n$ and the degree of the polarization $\lambda$. The smallest…

Let $(A,\Theta)$ be a complex principally polarized abelian variety of dimension $g\geq 4$. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor $\Theta$ is irreducible,…

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…

In this paper we consider the singularities of the varieties parameterizing stable vector bundles of fixed rank and degree with sections on a smooth curve of genus at least two. In particular, we extend results of Y. Laszlo, and of the…

In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over…

The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no…

To every abelian subvariety of a principally polarized abelian variety $(A, \mathcal{L})$ we canonically associate a numerical class in the N\'eron-Severi group of $A$. We prove that these classes are characterized by their intersection…