Plane rational quartics and K3 surfaces

Abstract

We study actions of the symmetric group S₄ on K3 surfaces X that satisfy the following condition: there exists an equivariant birational contraction \(\bar r:X \to \bar X\) to a K3 surface \(\bar X\) with ADE singularities such that the quotient space \(\bar X\)/S₄ is isomorphic to P². We prove that up to smooth equivariant deformations there exist exactly 15 such actions of the group S₄ on K3 surfaces, and that these actions are realized as actions of the Galois groups on the Galoisations \(\bar X\) of the dualizing coverings of the plane which are associated with plane rational quartics without A₄, A₆, and E₆ singularities as their singular points.