Heights of hypersurfaces in toric varieties

Gualdi, Roberto

doi:10.2140/ant.2018.12.2403

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Heights of hypersurfaces in toric varieties

Roberto Gualdi

Vol. 12 (2018), No. 10, 2403–2443

DOI: 10.2140/ant.2018.12.2403

Abstract

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.

Keywords

toric variety, height of a variety, Ronkin function, Legendre–Fenchel duality, mixed integral

Mathematical Subject Classification 2010

Primary: 14M25

Secondary: 11G50, 14G40, 52A39

Milestones

Received: 14 November 2017

Revised: 16 July 2018

Accepted: 23 September 2018

Published: 1 February 2019

Authors

Roberto Gualdi
Institut de Mathématiques de Bordeaux Université de Bordeaux Talence France

Vol. 12, No. 10, 2018