Polyhedral adjunction theory

TL;DR

This paper introduces a combinatorial approach to adjunction theory for toric varieties, defining new geometric invariants of polytopes and establishing structural results with implications for Ehrhart theory and classification of polytopes.

Contribution

It defines the Q-codegree and nef value for polytopes, and proves a structure theorem linking high Q-codegree to lattice width one, bridging adjunction theory and polytope classification.

Findings

High Q-codegree implies lattice width one for polytopes.

Empty adjoint polytope P^(s) for s<2/(dim(P)+2) indicates lattice width one.

Results connect adjunction theory classification with lattice polytope classification.

Abstract

In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We define the adjoint polytope P^(s) as the set of those points in P, whose lattice distance to every facet of P is at least s. We prove a structure theorem for lattice polytopes P with high Q-codegree. If P^(s) is empty for some s < 2/(dim(P)+2), 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.