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

Okounkov bodies and restricted volumes along very general curves

“…Motivated by earlier works of Okounkov [Oko96,Oko03], Lazarsfeld and Mustat ¸ǎ gave an interesting construction in a recent paper [LM08], which associates a convex body ∆(D) ⊂ R d to any big divisor D on a projective variety X of dimension d. (Independent of [LM08], Kaveh and Khovanskii also came up with a similar construction around the same time: see [KK08,KK09].) This so called "Okounkov body" encodes many asymptotic invariants of the complete linear series |mD| as m goes to infinity.…”

Crystal bases and Newton–Okounkov bodies

“…In this section we consider valuations on the rings of sections of line bundles. We recall from [KaKh12a] the construction of a Newton-Okounkov body and the related results on Hilbert functions.…”

“…More precisely, we show that the string parametrization of a (dual) crystal basis (due to Littelmann [Litt98] and Berenstein-Zelevinsky [BeZe01]) extends to a natural geometric valuation on the field of rational functions C(G/B), constructed out of a coordinate system on a Bott-Samelson variety, where we regard the elements of the irreducible representation as polynomials on the open cell in G/B and hence rational functions on G/B. This interpretation of the string parametrization shows that the string polytopes associated to irreducible representations can be realized as Newton-Okounkov bodies for the flag variety of G. The notion of a Newton-Okounkov body is a far generalization of the notion of the Newton polytope of a toric variety (see [Ok96,Ok03,KaKh08,LaMu08,KaKh12a]). We believe that this opens new doors to study the fundamental notion of a crystal basis/canonical basis in representation theory and we expect it to make some properties of the crystal bases more transparent.…”

“…On the other hand following the pioneering works of A. Okounkov in [Ok96] and [Ok03], the author and A. G. Khovanskii ( [KaKh12a,KaKh08]), as well as Lazarsfeld and Mustata ( [LaMu08]), developed a theory of convex bodies associated to linear series on algebraic varieties. Let X be a d-dimensional projective algebraic variety with field of rational functions C(X).…”

“…The convex body ∆ has the property that the degree of L is given by d! times the volume of ∆(R(L)) (Theorem 1.15, see also [KaKh08,LaMu08,KaKh12a]).…”