Asymptotic base loci via Okounkov bodies

Abstract

Abstract. An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the size of simplexes contained in the Okounkov bodies.

Seshadri constants and Okounkov bodies revisited

“…As was observed in [KL3,Remark 4.9] and [CHPW,Example 7.4], the inequality ε(D; x) ≥ ξ(D; x) in Theorem 1.2 can be strict in general. Moreover, one can conclude from [CPW3,Remark 3.12] that it is impossible to extract the exact value of ε(D; x) from the set of Okounkov bodies.…”

“…Let f * m D = P m + N m be the divisorial Zariski decomposition. Since f −1 m (x) ∈ Supp(N m ), it follows from [CHPW,Lemma 3.9…”

“…See Section 2 for more details. In recent years, a considerable amount of research has been devoted to the study of the connection between local positivity of divisors and (infinitesimal) Okounkov bodies (see e.g., [CHPW,CPW1,CPW2,CPW3], [DKMS], [I], [KL1,KL2,KL3], [R]). It was explained in [CHPW], [KL1,KL2,KL3] that (inverted) standard simplices arise naturally in (infinitesimal) Okounkov bodies.…”

Local numerical equivalences and Okounkov bodies in higher dimensions

“…Based on these results, there have been extensive and thorough studies of asymptotic numerical positivity of divisors via Okounkov bodies. The recent results tell us that the "local" numerical properties such as moving Seshadri constant ε(||D||; x) can be computed from the Okounkov bodies ∆ Y• (D) by fixing Y n at a given point x of X (see [CHPW1], [KL1], [KL3]). One can also extract the "global" numerical properties such as ampleness, nefness, and the asymptotic base loci B + (D), B − (D) from the Okounkov bodies ∆ Y• (D) by varying admissible flags Y • (see [CHPW1], [KL1], [KL2], [KL3]).…”

Global geometry of surfaces defined by non-positive and negative at infinity valuations

“…Many authors are interested in these convex sets since they also reveal interesting information about invariants and positivity properties of divisors on X [15,104,81,79,80,82]. Other recent works about Newton-Okounkov bodies are [26,27,92,12,101].…”