On base point freeness  in positive characteristic

Cascini, Paolo; Xu, Hiromu Tanaka Chenyang

doi:10.24033/asens.2269

Geodesic

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On base point freeness in positive characteristic
[Sur la vacuité du lieu-base en caractéristique positive]

Cascini, Paolo ; Xu, Hiromu Tanaka Chenyang

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1239-1272.

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Abstract (VO)
Résumé

We prove that if $(X, A + B)$ is a pair defined over an algebraically closed field of positive characteristic such that $(X, B)$ is strongly $F$ -regular, $A$ is ample and $K_{X} + A + B$ is strictly nef, then $K_{X} + A + B$ is ample. Similarly, we prove that for a log pair $(X, A + B)$ with $A$ being ample and $B$ effective, $K_{X} + A + B$ is big if it is nef and of maximal nef dimension. As an application, we establish a rationality theorem for the nef threshold and various results towards the minimal model program in dimension three in positive characteristic.

Nous démontrons que, si $(X, A + B)$ est une paire définie sur un corps algébriquement clos de caractéristique positive telle que $(X, B)$ est fortement $F$ -régulière, $A$ est ample et $K_{X} + A + B$ est strictement nef, alors $K_{X} + A + B$ est ample. De la même manière, nous prouvons que, si $(X, A + B)$ est une paire telle que $A$ est ample et $B$ est grand (« big »), alors une condition nécessaire et suffisante pour que le diviseur $K_{X} + A + B$ soit grand est qu'il soit nef et de dimension nef maximale. Nous utilisons ces résultats pour démontrer un théorème de rationalité pour le seuil nef, ainsi que plusieurs résultats nécessaires au programme des modèles minimaux en caractéristique positive en dimension trois.

Publié le : 2015-05-27

MR Zbl

DOI : 10.24033/asens.2269

Classification : 14E30, 13A35.
Keywords: Birational geometry, positive characteristic.
Mots-clés : Géométrie birationnele, caractéristique positive.

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     author = {Cascini, Paolo and Xu, Hiromu Tanaka Chenyang},
     title = {On base point freeness  in positive characteristic},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1239--1272},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {5},
     year = {2015},
     doi = {10.24033/asens.2269},
     mrnumber = {3429479},
     zbl = {1408.14020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2269/}
}

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Cascini, Paolo; Xu, Hiromu Tanaka Chenyang. On base point freeness  in positive characteristic. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1239-1272. doi : 10.24033/asens.2269. http://geodesic.mathdoc.fr/articles/10.24033/asens.2269/

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