K3 polytopes and their quartic surfaces

Tropical geometry is a recent area of mathematics at the interface between algebraic and polyhedral geometry. Tropical varieties, the tropical counterpart of algebraic varieties, are polyhedral complexes satisfying certain combinatorial properties. The closure of the connected components of the complement of a tropical hypersurface are called regions. They have the structure of convex polyhedra. A $3$-dimensional polytope is a K3 polytope if it is the closure of the bounded region of a smooth tropical quartic surface.

In this talk we begin by studying properties of K3 polytopes. In particular we exploit their duality to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Then we focus on quartic surfaces that tropicalize to K3 polytopes, and we look at them through the lenses of Geometric Invariant Theory. These are first steps on the computational study of tropical quartic surfaces.

In the presentation we will highlight computational aspects and further questions related to the project. The talk is based on joint work with Gabriele Balletti and Bernd Sturmfels.