arXiv subjects

Thomas Bauer

Publications and source records attributed to Thomas Bauer.

57 records · Page 4Linked to original sources

arXiv1998

Cyclic coverings and higher order embeddings of algebraic varieties

An algebraic variety X is embedded to the order k via a line bundle L if the global sections of L generate all (simultaneous) jets of order k on X or if they separate all zero-dimensional subschemes of length at most k+1. Even though we refer to both situations as "higher order embeddings", the first notion (in which case L is said to be k-jet ample) is stronger than the second one (when L is k-very ample). The purpose of this paper is to study higher order embeddings of cyclic coverings π:Y\to X via line bundles given by pulling back "sufficiently positive" line bundles on X. Given a line bundle L on X, we relate the order of the embedding defined by π^*L to that of L and of certain rank 1 summands of the vector bundle L\tensorπ_*\calo_Y. As expected, the sufficient conditions for π^*L to be k-jet ample are stronger then the ones needed in order for π^*L to be k-very ample.

math.AG↗

arXiv1998

Seshadri constants and periods of polarized abelian varieties

Consider a polarized abelian variety $(A,L)$ over the field of complex numbers. Following Demailly, one can associate to $(A,L)$ a real number $ε(A,L)$, its {\em Seshadri constant}, which in effect measures how much of the positivity of $L$ can be concentrated at any given point of $A$. There has been considerable recent interest in finding bounds on the Seshadri constants of abelian varieties and on smooth projective varieties in general. In the present paper we first generalize an approach of Buser and Sarnak to give a lower bound on the minimal period length of $(A,L)$ in terms of the type of the polarization, which by a recent result of Lazarsfeld leads to a lower bound on the Seshadri constant of $(A,L)$. Secondly, we consider Prym varieties and show that they have small Seshadri constants and therefore unusually small periods. Finally, we obtain refined results for the case of abelian surfaces, which imply in particular the surprising fact that Seshadri constants are always rational in this case.

math.AG↗

arXiv1997

On the cone of curves of an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample, however, it is in general difficult to determine the structure of $\bar NE(X)$. The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the abelian varieties $X$ are determined such that the closed cone $\bar NE(X)$ is rational polyhedral. The result can also be formulated in terms of the nef cone of $X$ or in terms of the semi-group of effective classes in the Néron-Severi group of $X$.

alg-geom↗