Tropical Compactifications

We study compactifications of very affine varieties defined by imposing a polyhedral structure on the non-archimedean amoeba. These com-pactifications have divisorial boundary with combinatorial normal croosings. We consider some examples including M 0,n ⊂ M 0,n (and more generally log canonical models of complements of hyperplane arrangements) and tropical recompactifications of Chow quotients of Grassmannians. §1. Introduction 1.1. Very Affine Varieties. Let X be a connected affine algebraic scheme such that O(X) is generated by its units O * (X). In this case we say that X is a very affine scheme. X is a closed subscheme in its intrinsic torus T = Hom(Λ, k *), where Λ = O * (X)/k * is a finitely generated free Z-module. The embedding X ⊂ T is unique up to a translation by an element of T. A category V is defined as follows: objects of V are pairs (X, P), where X is a very affine scheme and P is a normal toric variety of T. We do not require that T ⊂ P. A morphism in V is a regular morphism of toric varieties extending a regular morphism of very affine varieties (the association X → T is covariant by pullback).