Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

• Autores: Kiumars Kaveh, A. G. Khovanskii
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 176, Nº 2, 2012, págs. 925-978
• Idioma: inglés
• DOI: 10.4007/annals.2012.176.2.5
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• Resumen

  • Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras and linear series on varieties. We prove that any semigroup in the lattice Z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results. We show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of the Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.