Polyhedral adjunction theory

Di Rocco, Sandra; Haase, Christian; Nill, Benjamin; Paffenholz, Andreas

doi:10.2140/ant.2013.7.2417

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Polyhedral adjunction theory

Sandra Di Rocco, Christian Haase, Benjamin Nill and Andreas Paffenholz

Vol. 7 (2013), No. 10, 2417–2446

DOI: 10.2140/ant.2013.7.2417

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 $ℚ$ -codegree and the nef value of a rational polytope $P$ . We prove a structure theorem for lattice polytopes $P$ with large $ℚ$ -codegree. For this, 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 $s$ . It follows from our main result that if $P^{(s)}$ is empty for some $s < 2 ∕ (d i m P + 2)$ , 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.

Keywords

convex polytopes, toric varieties, adjunction theory

Mathematical Subject Classification 2010

Primary: 14C20

Secondary: 14M25, 52B20

Milestones

Received: 27 June 2012

Revised: 2 November 2012

Accepted: 16 March 2013

Published: 18 January 2014

Authors

Sandra Di Rocco
Department of Mathematics KTH SE-10044 Stockholm Sweden

Christian Haase
Institut für Mathematik Goethe-Universität Frankfurt Robert-Mayer-Str. 10 D-60325 Frankfurt Germany

Benjamin Nill
Department of Mathematics Stockholm University SE 106 91 Stockholm Sweden

Andreas Paffenholz
Fachbereich Mathematik Technische Universität Darmstadt Dolivostr. 15 D-64293 Darmstadt Germany

Vol. 7, No. 10, 2013