On the integrality of Seshadri constants of abelian surfaces

Abstract

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 $${\mathbb {Z}}[i]$$ or in $${\mathbb {Z}}[(1\,{+}\,i\sqrt{3})/2]$$ or if E has no complex multiplication at all, then it is known that for every ample line bundle L on , the Seshadri constant $$\varepsilon (L)$$ 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.