On base point freeness in positive characteristic

Cascini, Paolo; Xu, Hiromu Tanaka Chenyang

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

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.

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.

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

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