Open Access
September 2016 On the anti-canonical geometry of $\mathbb{Q}$-Fano threefolds
Meng Chen, Chen Jiang
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J. Differential Geom. 104(1): 59-109 (September 2016). DOI: 10.4310/jdg/1473186539


For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $\lvert -mK_X \rvert$ and prove that the anti-$m$-canonical map $\varphi - m$ is birational onto its image for all $m \geq 39$. By a weak $\mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $\mathbb{Q}$-Cartier, nef and big. For weak $\mathbb{Q}$-Fano 3-folds, we prove that $\varphi - m$ is birational onto its image for all $m \geq 97$.


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Meng Chen. Chen Jiang. "On the anti-canonical geometry of $\mathbb{Q}$-Fano threefolds." J. Differential Geom. 104 (1) 59 - 109, September 2016.


Published: September 2016
First available in Project Euclid: 6 September 2016

zbMATH: 1375.14137
MathSciNet: MR3544286
Digital Object Identifier: 10.4310/jdg/1473186539

Rights: Copyright © 2016 Lehigh University

Vol.104 • No. 1 • September 2016
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