Polyhedral adjunction theory

Abstract

In this paper we offer a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we explore two convex-geometric notions: the $\mathbb{Q}$-codegree and the nef value of a rational polytope $P$. We prove a structure theorem for lattice polytopes $P$ with large $\mathbb{Q}$-codegree. For this, we define the adjoint polytope $P^{\left(s\right)}$ as the set of those points in $P$ whose lattice distance to every facet of $P$ is at least $s$. It follows from our main result that if $P^{\left(s\right)}$ is empty for some $s<2∕\left(dimP+2\right)$, then the lattice polytope $P$ has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.