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On K3 Surface Quotients of K3 or AbelianSurfaces

Published online by Cambridge University Press:  20 November 2018

Alice Garbagnati
Alice Garbagnati*
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italia e-mail: [email protected]
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Abstract

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The aim of this paper is to prove that a $\text{K3}$ surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a $\text{K3}$ surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the $\text{K3}$ surfaces that are (rationally) $G$-covered by Abelian or $\text{K3}$ surfaces (in the latter case $G$ is an Abelian group). When $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result is already known; we extend it to the other cases.

Moreover, we prove that a $\text{K3}$ surface ${{X}_{G}}$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on ${{X}_{G}}$. Again, this result was known only in some special cases, in particular, if $G$ has order 2 or 3.

Keywords

14J2814J5014J10K3 surfacesKummer surfacesKummer type latticequotient surfaces
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 2 , 01 April 2017 , pp. 338 - 372
Copyright
Copyright © Canadian Mathematical Society 2017

References

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