On the anti-canonical geometry of weak -Fano threefolds II
Annales de l'Institut Fourier, Volume 70 (2020) no. 6, pp. 2473-2542.

By a canonical (resp. terminal) weak -Fano 3-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is -Cartier, nef and big. For a canonical weak -Fano 3-fold V, we show that there exists a terminal weak -Fano 3-fold X, being birational to V, such that the m-th anti-canonical map defined by |-mK X | is birational for all m52. As an intermediate result, we show that for any K-Mori fiber space Y of a canonical weak -Fano 3-fold, the m-th anti-canonical map defined by |-mK Y | is birational for all m52.

Par variété de dimension trois canonique (resp. terminale) faiblement -Fano, nous entendons une variété normale projective avec au plus des singularités canoniques (resp. terminales) sur laquelle le diviseur anticanonique est -Cartier, nef et big. Pour une variété de dimension trois canonique faiblement -Fano V, nous montrons qu’il existe une variété de dimension trois terminale faiblement -Fano X birationnelle à V, de sorte que l’application pluri-anticanonique définie par |-mK X | est birationnelle sur son image pour tous les m52. Comme un résultat intermédiaire, nous montrons que pour n’importe quelle K-fibration de Mori Y d’une variété de dimension trois canonique faiblement -Fano, l’application pluri-anticanonique définie par |-mK Y | est birationnelle sur son image pour tous les m52.

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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

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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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, Volume 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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