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Newton convex bodies, semigroups of integral points, graded algebras and intersection theory | | Kiumars Kaveh
; A.G. Khovanskii
; | | Date: |
21 Apr 2009 | | Abstract: | Generalizing the notion of Newton polyhedron, we define the Newton convex
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 wide 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 implies a generalization of it for arbitrary
linear series. Applications to intersection theory include a far-reaching
generalization of Kushnirenko theorem (from Newton polyhedra theory) and a new
version of Hodge inequality. We also give elementary proofs of
Alexandrov-Fenchel inequality (and its corollaries) in convex geometry and
their analogues in algebraic geometry. | | Source: | arXiv, 0904.3350 | | Services: | Forum | Review | PDF | Favorites | |
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