In
algebraic geometry, a Newton–Okounkov body, also called an Okounkov body, is a
convex body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior.
A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
associated to a
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
(or more generally a linear system) on a
variety
Variety may refer to:
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* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
. The convex geometry of a Newton–Okounkov body encodes (asymptotic) information about the geometry of the variety and the divisor. It is a large generalization of the notion of the
Newton polytope In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered negligible relative to the others. Specifically, give ...
of a projective
toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be nor ...
.
It was introduced (in passing) by
Andrei Okounkov
Andrei Yuryevich Okounkov (russian: Андре́й Ю́рьевич Окунько́в, ''Andrej Okun'kov'') (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematic ...
in his papers in the late 1990s and early 2000s. Okounkov's construction relies on an earlier result of
Askold Khovanskii
Askold Georgievich Khovanskii (russian: Аскольд Георгиевич Хованский; born 3 June 1947, Moscow) is a Russian and Canadian mathematician currently a professor of mathematics at the University of Toronto, Canada. His area ...
on semigroups of lattice points. Later, Okounkov's construction was generalized and systematically developed in the papers of
Robert Lazarsfeld and Mircea Mustață as well as Kiumars Kaveh and Khovanskii.
Beside Newton polytopes of toric varieties, several polytopes appearing in representation theory (such as the
Gelfand–Zetlin polytopes and the string polytopes of Peter Littelmann and Arkady Berenstein–
Andrei Zelevinsky
Andrei Vladlenovich Zelevinsky (; 30 January 1953 – 10 April 2013) was a Russian-American mathematician who made important contributions to algebra, combinatorics, and representation theory, among other areas.
Biography
Zelevinsky graduated in ...
) can be realized as special cases of Newton–Okounkov bodies.
References
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External links
Oberwolfach workshop "Okounkov bodies and applications"
BIRS workshop "Positivity of linear series and vector bundles"
BIRS workshop "Convex bodies and representation theory"
Oberwolfach workshop "New developments in Newton–Okounkov bodies"
{{DEFAULTSORT:Newton-Okounkov body
Algebraic geometry
Multi-dimensional geometry