The Nash problem for a toric pair and the minimal log-discrepancy

Shihoko Ishii

doi:10.1016/j.crma.2010.07.034

Algebraic Geometry

The Nash problem for a toric pair and the minimal log-discrepancy
[Problème de Nash pour une paire torique et la log-discrépance minimale]

Présenté par : Bernard Malgrange

Shihoko Ishii ¹

¹ Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, 152-8551 Tokyo, Japan

Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 985-988.

Résumé
Abstract

Dans cette Note, nous formulons le problème de Nash pour une paire constituée d'une variété torique et d'un idéal invariant. Nous montrons que le problème admet une réponse positive. Nous montrons aussi que la log-discrépance minimale, si elle est finie, est calculée par un diviseur correspondant à une composante de Nash. D'autre part, si la log-discrépance minimale est −∞, alors il existe une composante de Nash dont le diviseur correspondant est de log-discrépance négative.

This Note formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepancy is computed by a divisor corresponding to a Nash component, if the minimal log-discrepancy is finite. On the other hand there exists a Nash component such that the corresponding divisor has negative log-discrepancy, if the minimal log-discrepancy is −∞.

Reçu le : 2010-03-25
Accepté le : 2010-07-27
Publié le : 2010-09-01

DOI : 10.1016/j.crma.2010.07.034

Affiliations des auteurs :

Shihoko Ishii ¹

¹ Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, 152-8551 Tokyo, Japan

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Shihoko Ishii. The Nash problem for a toric pair and the minimal log-discrepancy. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 985-988. doi : 10.1016/j.crma.2010.07.034. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.034/

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