Tropical Kummer quartic surfaces

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.