On the integrality of Seshadri constants of abelian surfaces

TL;DR

This paper investigates when Seshadri constants on abelian surfaces are integers, revealing that only certain elliptic curves with specific complex multiplication properties exhibit this integrality, and explores the connection between integrality and elliptic curves.

Contribution

It characterizes elliptic curves on abelian surfaces for which Seshadri constants are integers and examines the relationship between integrality and elliptic curves.

Findings

Only elliptic curves with specific complex multiplication have integer Seshadri constants.

On any abelian surface, integrality of Seshadri constants relates to the presence of elliptic curves.

The paper provides a complete classification for self-products of elliptic curves regarding this property.

Abstract

In this paper we consider the question of when Seshadri constants on abelian surfaces are integers. Our first result concerns self-products $E\times E$ of elliptic curves: If $E$ has complex multiplication in $\Z[i]$ or in $\Z[\frac12(1+i\sqrt3)]$ or if $E$ has no complex multiplication at all, then it is known that for every ample line bundle $L$ on $E\times E$, the Seshadri constant $\eps(L)$ is an integer. We show that, contrary to what one might expect, these are in fact the only elliptic curves for which this integrality statement holds. Our second result answers the question how -- on any abelian surface~-- integrality of Seshadri constants is related to elliptic curves.