#### 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