\(K3\)-surfaces with special symmetry (Q2876098)

In this dissertation classification problems for \(K3\)-surfaces with finite group actions are considered. Special emphasis is put on \(K3\)-surfaces with antisymplectic involutions and compatible actions of symplectic transformations. Given a finite group \(G\) it is investigated if it can act in a compatible fashion on a \(K3\)-surfaces \(X\) with antisymplectic involution and to what extend the action of \(G\) determines the geometry of \(X\). If \(G\) has large order or rich structure a precise description of \(X\) as a branched double cover of a Del Pezzo surface is obtained. In particular, this leads to classification results for the complete list of eleven maximal finite groups of symplectic automorphisms of \(K3\)-surfaces. The remaining part of the thesis deals with generalizations to non-maximal groups and K3-surfaces with non-symplectic automorphisms of higher order. The classification relies on an equivariant minimal model program for surfaces, the study of finite group actions on Del Pezzo surfaces, and the combinatorial intersection geometry of naturally associated exceptional and fixed point curves.