Convex bodies associated to linear series

Abstract

4 e série, t. 42, 2009, p. 783 à 835 CONVEX BODIES ASSOCIATED TO LINEAR SERIES ʙʏ Rʙʀ LAZARSFELD ɴ Mɪʀ MUSTAT ,Ȃ Aʙʀ. -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 such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give some applications and examples.R. -Dans son travail sur la log-concavité des multiplicités, Okounkov montre au passage que l'on peut associer un corps convexe à un système linéaire sur une variété projective, puis utiliser la géométrie convexe pour étudier ces systèmes linéaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibrés en droites amples, il se trouve que sa construction s'étend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses propriétés de base des invariants asymptotiques des systèmes linéaires, et ouvre la porte à de nombreuses extensions. Le but de cet article est d'initier un développement systématique de la théorie et de donner quelques applications et exemples.

Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

“…point x) computed with respect to K. This means that when computing α, we use the height H L relative to K and 12. Suppose x ∈ X(k), L a line bundle on X, and {x i } → x a sequence of points in X(k) approximating x.…”

Asymptotic base loci via Okounkov bodies

“…The following is a generalization of [LM,Theorem 4.26] and [J,Theorem 3.4], and it will play a crucial role in studying the augmented base loci and moving Seshadri constants. …”

“…The Zariski decomposition plays a crucial role in computing the Okounkov body of a big divisor in the surface case (see [LM,Theorem 6.4]). As before, X is a smooth projective variety of dimension n.…”

“…It is too much to expect to obtain the exact values of the moving Seshadri constants by only considering the Okounkov bodies on X. On the other hand, one can obtain the exact values by using the infinitesimal Okounkov bodies (see [LM,Remark 5.5], [KL3,Theorem C]). However, computing the infinitesimal Okounkov bodies is quite difficult in general.…”

“…Let D be a divisor on a smooth projective variety X of dimension n. Then the Okounkov body ∆ Y• (D) is a convex subset of the Euclidean space R n associated to D with respect to an admissible flag Y • . In [CHPW], we defined and studied the valuative Okounkov It was shown that two pseudoeffective divisors are numerically equivalent to each other if and only if the associated limiting Okounkov bodies with respect to all admissible flags coincide ( [LM,Proposition 4.1], [J, Theorem A], [CHPW, Theorem C]). We also refer to [R] for a recent study on the local positivity and the local numerical equivalence using Okounkov bodies in the surface case.…”