# Seminar in Real and Complex Geometry

## Plane rational quartics and $K3$ surfaces

Abstract

 The talk is devoted to the actions of the symmetric group $\mathbb S_4$ on $K3$ surfaces $X$ satisfying the following properties: \begin{itemize} \item[$(*)$] the action of the alternating group $\mathbb A_4\subset \mathbb S_4$ on $X$ is symplectic; \item[$(**)$] there is an equivariant birational contraction $r: X\to \overline X$ to a surface $\overline X$ with $A-D-E$-singularities such that $\overline X/\mathbb S_4\simeq \mathbb P^2$. \end{itemize} It will be explained that up to smooth equivariant deformations there are exactly 16 such actions of $\mathbb S_4$ on the $K3$ surfaces and all these actions are realised as the actions of the Galois group on the Galois closures $\overline X$ of the dualising coverings of the projective plane associated with plane rational quartics having no the singularities of the types $A_4$, $A_6$, and $E_6$.