K3 Polytopes and their Quartic Surfaces

TL;DR

This paper classifies K3 polytopes with up to 30 vertices, explores their combinatorial properties, and studies the associated tropical quartic surfaces, revealing their stability and singular loci.

Contribution

It provides a comprehensive classification of K3 polytopes up to 30 vertices and analyzes their relation to tropical quartic surfaces and stability properties.

Findings

Number of K3 polytopes with up to 30 vertices is 36,297,333

K3 polytopes are dual to unimodular triangulations of reflexive polytopes

Tropical quartic surfaces associated with K3 polytopes are stable and have specific singular loci

Abstract

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to $30$ vertices. Their number is $36\,297\,333$. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.