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- W2145758416 abstract "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 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." @default.
- W2145758416 created "2016-06-24" @default.
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- W2145758416 date "2009-04-21" @default.
- W2145758416 modified "2023-09-23" @default.
- W2145758416 title "Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory" @default.
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- W2145758416 doi "https://doi.org/10.48550/arxiv.0904.3350" @default.
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