Vol. 12, No. 10, 2018

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

Volume 15
Issue 5, 1077–1342
Issue 4, 821–1076
Issue 3, 569–820
Issue 2, 309–567
Issue 1, 1–308

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

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
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Heights of hypersurfaces in toric varieties

Roberto Gualdi

Vol. 12 (2018), No. 10, 2403–2443

For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the v-adic roof functions associated to the metric and the Legendre–Fenchel dual of the v-adic Ronkin function of the Laurent polynomial of the cycle.

toric variety, height of a variety, Ronkin function, Legendre–Fenchel duality, mixed integral
Mathematical Subject Classification 2010
Primary: 14M25
Secondary: 11G50, 14G40, 52A39
Received: 14 November 2017
Revised: 16 July 2018
Accepted: 23 September 2018
Published: 1 February 2019
Roberto Gualdi
Institut de Mathématiques de Bordeaux
Université de Bordeaux