Tropical Kummer quartic surfaces

TL;DR

This paper introduces tropical Kummer quartic surfaces as images of tropical abelian surfaces, explores their properties, and establishes their relation to classical Kummer surfaces and skeletons in nonarchimedean geometry.

Contribution

It defines tropical Kummer surfaces via tropical theta functions, introduces rational polyhedral orbifolds, and connects tropical and classical Kummer surfaces through faithful tropicalizations.

Findings

Tropical Kummer surfaces are included in tropicalizations of classical Kummer surfaces.

Faithful embeddings of tropical Kummer surfaces as rational polyhedral orbifolds.

Canonical skeletons of certain Kummer surfaces coincide with Kontsevich-Soibelman skeletons.

Abstract

We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties, showing that they are included in the tropicalizations of Kummer quartic surfaces defined over nonarchimdean valued fields. In the course of this work, we introduce the notion of a rational polyhedral orbifold and we provide faithful embeddings of tropical Kummer surfaces as such. Further, we show faithful tropicalizations of the canonical skeletons of certain Kummer surfaces over nonarchimdean valued fields. Under a suitable assumption on the base field, the canonical skeletons coincide with the Kontsevich--Soibelman skeletons.