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On the integrality of Seshadri constants of abelian surfaces

  • Thomas BauerEmail author
  • Felix Fritz Grimm
  • Maximilian Schmidt
Research Article

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 Open image in new window , 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.

Keywords

Abelian surface Elliptic curve Complex multiplication Seshadri constant Integral 

Mathematics Subject Classification

14C20 14H52 14Jxx 14Kxx 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

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