Heights of hypersurfaces in toric varieties

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.

arXiv comment39 pages. In version 2: Theorem 1.6 rephrased under weaker assumptions, example of binomial hypersurfaces and related results added; other minor changes. Accepted for publication in "Algebra & Number Theory"

References in corpus (6)

• Formes différentielles réelles et courants sur les espaces de Berkovich
• Tropical varieties for non-archimedean analytic spaces
• Mesures et équidistribution sur les espaces de Berkovich
• A tropical approach to non-archimedean Arakelov theory
• Mahler measures and computations with regulators
• Arithmetic geometry of toric varieties. Metrics, measures and heights

Cited by in corpus (1)

• On the Northcott property for special values of L-functions