On the integrality of Seshadri constants of abelian surfaces
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2018
- DOI:
- 10.48550/arXiv.1805.05413
- arXiv:
- arXiv:1805.05413
- Bibcode:
- 2018arXiv180505413B
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14C20;
- 14H52;
- 14Jxx;
- 14Kxx