On the $K3$ surface with $\mathfrak{S}_4 \times \mathfrak{S}_4$ action

TL;DR

This work constructs an explicit polarized K3 surface with a faithful $\\mathfrak{S}_4\\times\\mathfrak{S}_4$-action, situating it within the Brandhorst--Hashimoto classification of non-maximal-symplectic extensions and showing its uniqueness. The authors provide a concrete projective model in $\\mathbf{P}^1\\times\\mathbf{P}^1\\times\\mathbf{P}^1\\times\\mathbf{P}^1$ and establish isomorphisms to Schur's quartic, the T_{192}-based model, and a Kummer surface, with transcendental lattice $8448$ and a symplectic subgroup isomorphic to $\\mathfrak{A}_{4,4}$. They further relate these models by explicit birational maps in characteristic zero and describe the intersection of polarization-preserving automorphism groups, finding it to be $C_4\\rtimes\\mathfrak{S}_4$ with symplectic part $C_2\\times\\mathfrak{S}_4$. The results yield explicit geometric realizations across characteristics and sharpen the understanding of how these highly symmetric K3 surfaces relate to known extremal examples.

Abstract

By a lattice theoretic approach, Brandhorst--Hashimoto has made the list of K3 surfaces with finite groups of automorphisms which properly contain a maximal symplectic automorphism group. We give $3$ different explicit descriptions to the $K3$ surface with an action of $\mathfrak{S}_4\times \mathfrak{S}_4$, with various characterizations, and construct an explicit isomorphism to the Schur's quartic. We also calculate the intersection of the two polarization-preserving finite automorphism groups.

Theorems & Definitions (23)

• Theorem 1.1: main theorem
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• Theorem 2.3: shioda1977singular
• Theorem 2.4: Huybrechts_2016
• Theorem 2.5: DolgKeum,mukai1988finite
• Lemma 3.1
• proof
• Proposition 3.2