Vol. 12, No. 10, 2018

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Heights of hypersurfaces in toric varieties

Roberto Gualdi

Vol. 12 (2018), No. 10, 2403–2443
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