Existence of flips and minimal models  for 3-folds in char $p$

Birkar, Caucher

doi:10.24033/asens.2279

Existence of flips and minimal models for 3-folds in char

p

[Existence de flips et de modèles minimaux pour les variétés de dimension 3 en caractéristique

p

]

Birkar, Caucher

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 169-212

Abstract (VO)
Résumé

We will prove the following results for 3-fold pairs $(X, B)$ over an algebraically closed field $k$ of characteristic $p > 5$ : log flips exist for $ℚ$ -factorial dlt pairs $(X, B)$ ; log minimal models exist for projective klt pairs $(X, B)$ with pseudo-effective $K_{X} + B$ ; the log canonical ring $R (K_{X} + B)$ is finitely generated for projective klt pairs $(X, B)$ when $K_{X} + B$ is a big $ℚ$ -divisor; semi-ampleness holds for a nef and big $ℚ$ -divisor $D$ if $D - (K_{X} + B)$ is nef and big and $(X, B)$ is projective klt; $ℚ$ -factorial dlt models exist for lc pairs $(X, B)$ ; terminal models exist for klt pairs $(X, B)$ ; ACC holds for lc thresholds, etc.

Étant donnée une paire $(X, B)$ de dimension trois sur un corps algébriquement clos $k$ de caractéristique $p > 5$ , nous prouvons les résultats suivants : existence de log-flips lorsque la paire est $ℚ$ -factorielle et dlt; existence de log-modèles minimaux lorsque la paire est klt, projective, et avec $K_{X} + B$ pseudo-effectif ; finitude de l'anneau log-canonique $R (K_{X} + B)$ lorsque la paire est klt, projective, et avec $K_{X} + B$ gros; semi-amplitude pour un $ℚ$ -diviseur nef et gros $D$ , sous la condition que $D - (K_{X} + B)$ est nef et gros et que $(X, B)$ est klt et projective; existence de modèles dlt et $ℚ$ -factoriels lorsque la paire est lc; existence de modèles terminaux lorsque la paire est klt; validité de la Conjecture ACC pour le seuil lc, etc.

Publié le : 2016-07-21

MR Zbl

DOI : 10.24033/asens.2279

Classification : 14E30
Keywords: Flip, minimal model, log canonical ring, log canonical threshold, characteristic $p$.
Mots-clés : Flip, modèles minimaux, anneau log-canonique, seuil log-canonique, caractéristique $p$.

@article{ASENS_2016__49_1_169_0,
     author = {Birkar, Caucher},
     title = {Existence of flips and minimal models  for 3-folds in char~$p$
            },
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {169--212},
     year = {2016},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {1},
     doi = {10.24033/asens.2279},
     mrnumber = {3465979},
     zbl = {1346.14040},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2279/}
}

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Birkar, Caucher. Existence of flips and minimal models  for 3-folds in char $p$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 169-212. doi: 10.24033/asens.2279

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