Convex bodies associated to linear series

Robert Lazarsfeld; Mircea Mustață

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 5, page 783-835
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Lazarsfeld, Robert, and Mustață, Mircea. "Convex bodies associated to linear series." Annales scientifiques de l'École Normale Supérieure 42.5 (2009): 783-835. <http://eudml.org/doc/272239>.

@article{Lazarsfeld2009,
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.},
author = {Lazarsfeld, Robert, Mustață, Mircea},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {algebraic varieties; linear series; convex bodies},
language = {eng},
number = {5},
pages = {783-835},
publisher = {Société mathématique de France},
title = {Convex bodies associated to linear series},
url = {http://eudml.org/doc/272239},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Lazarsfeld, Robert
AU - Mustață, Mircea
TI - Convex bodies associated to linear series
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 5
SP - 783
EP - 835
AB - 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.
LA - eng
KW - algebraic varieties; linear series; convex bodies
UR - http://eudml.org/doc/272239
ER -

References

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