Vol. 7, No. 10, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Polyhedral adjunction theory

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

Vol. 7 (2013), No. 10, 2417–2446
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 -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.

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