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

TL;DR

This paper develops an algorithm to compute G-invariant projective varieties and applies it to construct explicit models of K3 surfaces with specific symmetry groups, including a K3 surface with an order 192*2 automorphism.

Contribution

It introduces a novel algorithm for computing G-invariant subvarieties and uses it to explicitly construct K3 surfaces with complex symmetry groups.

Findings

Constructed a projective model of a K3 surface with a T_{192}⋊μ_2 action.

Generated symmetric K3 surfaces with degree 8 polarization.

Demonstrated the effectiveness of the algorithm in algebraic geometry applications.

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 computing $G$-invariant projective varieties arising as an intersection of hypersurfaces of the same degree. We apply the algorithm to find a projective model of a polarized K3 surface with a faithful action of $T_{192}\rtimes \mu_2$ and some further symmetric K3 surfaces with a degree 8 polarization.