On the integrality of Seshadri constants of abelian surfaces

We consider the question of when Seshadri constants on abelian surfaces are integers. Our first result concerns self-products E × E of elliptic curves: if E has complex multiplication in Z [ i ] or in Z [ ( 1 + i 3 ) / 2 ] or if E has no complex multiplication at all, then it is known that for every...

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Bibliographic Details
Published inEuropean journal of mathematics Vol. 6; no. 4; pp. 1264 - 1275
Main Authors Bauer, Thomas, Grimm, Felix Fritz, Schmidt, Maximilian
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2020
Subjects
Algebraic Geometry
Mathematics
Mathematics and Statistics
Research Article
Abelian surface
Seshadri constant
14Kxx
14C20
Complex multiplication
Integral
14H52
Elliptic curve
14Jxx
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Summary:We consider the question of when Seshadri constants on abelian surfaces are integers. Our first result concerns self-products E × E of elliptic curves: if E has complex multiplication in Z [ i ] or in Z [ ( 1 + i 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 ε ( 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.
ISSN:2199-675X
2199-6768
DOI:10.1007/s40879-019-00369-w