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Annales scientifiques de l'ENS - Parutions - série 4, 42 (2009)

Parutions < série 4, 42

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 42, fascicule 5 (2009)

Robert Lazarsfeld, Mircea Mustață
Convex bodies associated to linear series
Annales scientifiques de l'ENS 42, fascicule 5 (2009), 783-835

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Résumé :
Corps convexes associés à des séries linéaires
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.

Mots-clefs : Variétés algebriques, systèmes linéaires, corps convexes

Abstract:
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.

Keywords: Algebraic varieties, linear series, convex bodies

Class. math. : 14F05, 52C99


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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