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

@article{Kaveh2009NewtonOkounkovBS, title={Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory}, author={Kiumars Kaveh and Askold Khovanskii}, journal={arXiv: Algebraic Geometry}, year={2009} }

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…

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