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Heights of hypersurfaces in toric varieties. (English) Zbl 1410.14042

The notion of height of varieties generalizes the corresponding notion of height of points. It is a basic arithmetic invariant of a proper variety over \(\mathbb{Q}\). The article under review is devoted to the study of heights in toric varieties, a topic that has been treated in great generality by J. I. Burgos Gil et al. [Arithmetic geometry of toric varieties. Metrics, measures and heights. Paris: Société Mathématique de France (SMF) (2014; Zbl 1311.14050)].
Let \(\Sigma\) be a complete fan and let \(X_\Sigma\) be the corresponding proper toric variety of dimension \(n\) over \(\mathbb{Q}\). Denote by \(M\) the lattice of characters of the torus of \(X_\Sigma\), and denote by \(N\) its dual lattice. \(M_{\mathbb{R}}\) and \(N_{\mathbb{R}}\) will denote the corresponding real vector spaces. Let \(\mathfrak{M}\) stand for the set of places of \(\mathbb{Q}\).
The theory developed in [loc. cit.], provides a good amount of height functions, i.e. those that arise from toric line bundles equipped with toric metrics. Moreover, these height functions are described in terms of the underlying combinatorics of \(\Sigma\). The machinery developed there, is useful for computing heights of toric subvarieties but is not enough, for instance, to handle the case of a general cycle of codimension \(1\) in \(X_\Sigma\). This is exactly the main subject of the very well written work of Gualdi. The main result of this article is the following
Theorem 1. The height of \(Y\) with respect to \(\overline{D}\) is given by \[ h_{\overline{D}}(Y)=\sum_{v\in \mathfrak{M}}MI_{M_{\mathbb{R}}}(\vartheta_v,\ldots,\vartheta_v,\rho_{f,v}^\vee). \] Here \(Y\) is an irreducible hypersurface on \(X_\Sigma\) such that the generic point of \(Y\) lies in the dense open orbit of \(X_\Sigma\), which implies that \(Y\) can be described by an irreducible Laurent polynomial \(f\) with rational coefficients. By \(\overline{D}\) the author denotes a toric divisor on \(X_\Sigma\), equipped with an adelic semipositive metric which is invariant under the action of the torus of \(X_\Sigma\). Such a metric is associated with a family \(\{\vartheta_v\}_{v\in\mathfrak{M}}\) of continuous concave functions on the polytope associated to \(D\), such that, \(\vartheta_v=0\) for all but finitely many \(v\)’s. The description of this association is detailed in Section 3.
The function \(\rho_{f,v}\), defined for a fixed place \(v\) of \(\mathbb{Q}\), is called a \(v\)-adic Ronkin function of \(f\) and constitutes one of the key constructions of the article. It is related with the fibers of the tropicalization of \(f\), see Section 2 for the details. The notation \(\rho_{f,v}^\vee\) refers to the Legendre-Fenchel dual of \(\rho_{f,v}\), which is a concave function on \(M_\mathbb{R}\) supported on the Newton polytope of \(f\).
Finally, \(MI_{M_{\mathbb{R}}}\) denotes the mixed integral defined by Burgos Gil et al. [loc. cit.]. This is a multilinear symmetric real-valued function obtained from a suitably normalized Haar measure on \(M_\mathbb{R}\).
Furthermore, all the announced results are presented in the general adelic Arakelov framework, i.e. for an arbitrary base adelic field, see Section 3.
In developing the required tools for the proof of Theorem 1, the author also presents some extra results that are very welcome. They include some new properties of mixed integrals and a uniform study of \(v\)-adic Ronkin functions, in both, archimedean and nonarchimedean contexts. The author also describes combinatorially the Weil divisor of the rational function defined by a Laurent polynomial on a toric variety, see Section 4. At the end of the paper, Section 6, some well chosen examples are presented in order to relate the presented constructions with previous ones.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
11G50 Heights
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
52A39 Mixed volumes and related topics in convex geometry

Citations:

Zbl 1311.14050
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References:

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