Open Access
2013 Polyhedral adjunction theory
Sandra Di Rocco, Christian Haase, Benjamin Nill, Andreas Paffenholz
Algebra Number Theory 7(10): 2417-2446 (2013). DOI: 10.2140/ant.2013.7.2417


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.


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Sandra Di Rocco. Christian Haase. Benjamin Nill. Andreas Paffenholz. "Polyhedral adjunction theory." Algebra Number Theory 7 (10) 2417 - 2446, 2013.


Received: 27 June 2012; Revised: 2 November 2012; Accepted: 16 March 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1333.14010
MathSciNet: MR3194647
Digital Object Identifier: 10.2140/ant.2013.7.2417

Primary: 14C20
Secondary: 14M25 , 52B20

Keywords: adjunction theory , Convex polytopes , toric varieties

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 10 • 2013
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