Algebra & Number Theory

Heights of hypersurfaces in toric varieties

Roberto Gualdi

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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.

Article information

Algebra Number Theory, Volume 12, Number 10 (2018), 2403-2443.

Received: 14 November 2017
Revised: 16 July 2018
Accepted: 23 September 2018
First available in Project Euclid: 14 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 11G50: Heights [See also 14G40, 37P30] 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 52A39: Mixed volumes and related topics

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


Gualdi, Roberto. Heights of hypersurfaces in toric varieties. Algebra Number Theory 12 (2018), no. 10, 2403--2443. doi:10.2140/ant.2018.12.2403.

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