Vol. 7, No. 10, 2013

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Polyhedral adjunction theory

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

Vol. 7 (2013), No. 10, 2417–2446

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 -codegree and the nef value of a rational polytope P. We prove a structure theorem for lattice polytopes P with large -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(dimP + 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.

convex polytopes, toric varieties, adjunction theory
Mathematical Subject Classification 2010
Primary: 14C20
Secondary: 14M25, 52B20
Received: 27 June 2012
Revised: 2 November 2012
Accepted: 16 March 2013
Published: 18 January 2014
Sandra Di Rocco
Department of Mathematics
SE-10044 Stockholm
Christian Haase
Institut für Mathematik
Goethe-Universität Frankfurt
Robert-Mayer-Str. 10
D-60325 Frankfurt
Benjamin Nill
Department of Mathematics
Stockholm University
SE 106 91 Stockholm
Andreas Paffenholz
Fachbereich Mathematik
Technische Universität Darmstadt
Dolivostr. 15
D-64293 Darmstadt