Tropicalization of toric prevarieties

Abstract

The homogeneous spectrum of a multigraded finitely generated algebra (in the sense of Brenner and Schröer) always admits an embedding into a toric variety that is not necessarily separated, a so-called toric prevariety. In order to have a convenient framework to study the tropicalization of homogeneous spectra, we propose a tropicalization procedure for toric prevarieties and study its basic properties. With these tools at hand, we prove a generalization of Payne's and Foster–Gross–Payne's tropical limit theorem for divisorial schemes.