On the anti-canonical geometry of weak $\protect \mathbb{Q}$-Fano threefolds II

Chen, Meng; Jiang, Chen

doi:10.5802/aif.3367

On the anti-canonical geometry of weak

ℚ

-Fano threefolds II
[Sur la géométrie anticanonique des variétés faiblement

ℚ

-Fano de dimension trois II]

Chen, Meng ¹ ; Jiang, Chen ²

¹ School of Mathematical Sciences & Shanghai Centre for Mathematical Sciences Fudan University Shanghai 200433 (China)
² Shanghai Center for Mathematical Sciences Fudan University, Jiangwan Campus Shanghai 200438 (China)

Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2473-2542.

Résumé
Abstract

Par variété de dimension trois canonique (resp. terminale) faiblement $ℚ$ -Fano, nous entendons une variété normale projective avec au plus des singularités canoniques (resp. terminales) sur laquelle le diviseur anticanonique est $ℚ$ -Cartier, nef et big. Pour une variété de dimension trois canonique faiblement $ℚ$ -Fano $V$ , nous montrons qu’il existe une variété de dimension trois terminale faiblement $ℚ$ -Fano $X$ birationnelle à $V$ , de sorte que l’application pluri-anticanonique définie par $| - m K_{X} |$ est birationnelle sur son image pour tous les $m \geq 52$ . Comme un résultat intermédiaire, nous montrons que pour n’importe quelle $K$ -fibration de Mori $Y$ d’une variété de dimension trois canonique faiblement $ℚ$ -Fano, l’application pluri-anticanonique définie par $| - m K_{Y} |$ est birationnelle sur son image pour tous les $m \geq 52$ .

By a canonical (resp. terminal) weak $ℚ$ -Fano $3$ -fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $ℚ$ -Cartier, nef and big. For a canonical weak $ℚ$ -Fano $3$ -fold $V$ , we show that there exists a terminal weak $ℚ$ -Fano $3$ -fold $X$ , being birational to $V$ , such that the $m$ -th anti-canonical map defined by $| - m K_{X} |$ is birational for all $m \geq 52$ . As an intermediate result, we show that for any $K$ -Mori fiber space $Y$ of a canonical weak $ℚ$ -Fano $3$ -fold, the $m$ -th anti-canonical map defined by $| - m K_{Y} |$ is birational for all $m \geq 52$ .

Reçu le : 2019-02-16
Révisé le : 2019-09-18
Accepté le : 2019-11-20
Publié le : 2021-04-15

DOI : 10.5802/aif.3367

Classification : 14J45, 14J30, 14E30
Keywords: Fano variety, pluri-anti-canonical map
Mot clés : variétés de Fano, l’application pluri-anticanonique

Affiliations des auteurs :

Chen, Meng ¹ ; Jiang, Chen ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Chen, Meng and Jiang, Chen},
     title = {On the anti-canonical geometry of weak $\protect \mathbb{Q}${-Fano} threefolds {II}},
     journal = {Annales de l'Institut Fourier},
     pages = {2473--2542},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Chen, Meng; Jiang, Chen. On the anti-canonical geometry of weak $\protect \mathbb{Q}$-Fano threefolds II. Annales de l'Institut Fourier, Tome 70 (2020) no. 6, pp. 2473-2542. doi : 10.5802/aif.3367. https://aif.centre-mersenne.org/articles/10.5802/aif.3367/

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