Polyhedral adjunction theory

Details Polyhedral adjunction theory Polyhedral adjunction theory

2011-05-12

arxiv.org/abs/1105.2415v2

Mathematics

Combinatorics

24 pages, 8 figures

Scientific paper

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 with high Q-codegree. If P^(s) is empty for some s < 2/(dim(P)+2), then P has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. We remark that polyhedral adjunction theory even works in cases where the canonical divisor is not Q-Cartier. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

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