K3-surfaces with special symmetry: An example of classification by Mori-reduction

TL;DR

This paper classifies certain K3-surfaces with specific symmetries using equivariant Mori-reduction, providing a complete classification for a particular group and exploring related automorphism groups.

Contribution

It introduces an equivariant Mori-reduction approach to classify K3-surfaces with symmetries, exemplified by a complete classification for the group C3 ⋉ C7.

Findings

Complete classification for H = C3 ⋉ C7

Application to K3-surfaces with maximal symplectic automorphisms

Insights into automorphism groups like L2(7)

Abstract

The classification problem for K3-surfaces equipped with finite groups $H$ of symplectic symmetry centralized by an antisymplectic involution is considered. An approach via equivariant Mori-reduction is employed. This method, which has proved to be successful even for rather small groups, is exemplified here by giving a complete classification in the case $H = C_3 \ltimes C_7$. The consideration of this particular group is related to the study of K3-surfaces with maximal finite groups of symplectic automorphisms. Applications to the case $L_2(7)$ are given.