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A note on discrete mixed volume and Hodge-Deligne numbers. (English) Zbl 1409.14084

Summary: Generalizing the famous Bernstein-Kushnirenko theorem, A. G. Khovanskij proved in [Funct. Anal. Appl. 12, 38–46 (1978; Zbl 0406.14035)] a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskĭi in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14M10 Complete intersections
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Citations:

Zbl 0406.14035
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References:

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