On K3 Surface Quotients of K3 or AbelianSurfaces

Garbagnati, Alice

Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 338-372

Résumé

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.

Détail

DOI : 10.4153/CJM-2015-058-1

Mots-clés : 14J28, 14J50, 14J10, K3 surfaces, Kummer surfaces, Kummer type lattice, quotient surfaces