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… 
View PDF on arXiv
245 Citations
Highly Influential Citations
29
Background Citations
45
Methods Citations
34
Results Citations
9

245 Citations

Fano varieties with finitely generated semigroups in the Okounkov body construction
The Okounkov body is a construction which, to an effective divisor D on an n-dimensional algebraic variety X, associates a convex body in the n-dimensional Euclidean space R^n. It may be seen as a…
Convex bodies associated to actions of reductive groups
We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of…
Cohomology classes of complex approximable algebras
In [2], Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of…
Arithmetic Okounkov bodies and positivity of adelic Cartier divisors
. In algebraic geometry, theorems of K¨uronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov…
Reflexivity of Newton-Okounkov bodies of partial flag varieties
Assume that the valuation semigroup $\Gamma(\lambda)$ of an arbitrary partial flag variety corresponding to the line bundle $\mathcal L_\lambda$ constructed via a full-rank valuation is finitely…
Newton-Okounkov bodies and Segre classes
  • P. Aluffi
  • Mathematics
    American Journal of Mathematics
  • 2021
Given a homogeneous ideal in a polynomial ring over C, we adapt the construction of Newton-Okounkov bodies to obtain a convex subset of Euclidean space such that a suitable integral over this set…
Okounkov bodies of filtered linear series
Abstract We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the…
Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies
In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient…
Successive minima and asymptotic slopes in Arakelov Geometry
Let X be a normal and geometrically integral projective variety over a global field K and let D be an adelic Cartier divisor on X. We prove a conjecture of Chen, showing that the essential minimum…
Lifting Tropical Intersections
We show that points in the intersection of the tropicaliza- tions of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations…
...
...

References

SHOWING 1-10 OF 49 REFERENCES
Convex bodies associated to actions of reductive groups
We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of…
Okounkov bodies of filtered linear series
Abstract We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the…
Higher Dimensional Continued Fractions
The higher-dimensional analogue of a continuous fraction is the polyhedral surface, bounding the convex hull of the semigroup of the integer points in a simplicial cone of the euclidian space. The…
Convex bodies and algebraic equations on affine varieties
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a…
Newton polyhedron, Hilbert polynomial, and sums of finite sets
Theorem 1 is a special case of Theorem 2 of § 2. The polynomials in Theorems 1 and 2 are Hilbert polynomials of certain graded modules over the ring of polynomials in several variables. If a…
Convex bodies appearing as Okounkov bodies of divisors
NEWTON POLYTOPES FOR HOROSPHERICAL SPACES
A subgroup H of a reductive group G is horospherical if it contains a maximal unipotent subgroup. We describe the Grothendieck semigroup of invariant subspaces of regular functions on G=H as a semi-…
Convex Bodies Associated to Linear Series
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study…
Transforming metrics on a line bundle to the Okounkov body
LetL be a big holomorphic line bundle on a compact complex manifol dX.We show how to associate a convex function on the Okounkov body o f L t any continuous metrice onL.We will call this the…
Mixed volume and an extension of intersection theory of divisors
Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an…
...
...