Invariant Grassmannians and a K3 surface with an action of order 192*2

Abstract

Given a complex vector space V of finite dimension, its Grassmannian variety parametrizes all subspaces of V of a given dimension. Similarly, if a finite group G acts on V, its invariant Grassmannian parametrizes all the G−invariant subspaces of V of a given dimension. Based on this fact, we develop an algorithm for finding equations of G−invariant projective varieties arising as an intersection of hypersurfaces of the same degree.

We apply the algorithm to find equations describing a polarized K3 surface with a faithful action of $T_{192}⋊{\mu }_2$ and some further symmetric K3 surfaces with a degree 8 polarization.