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

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.