## Journal of Differential Geometry

- J. Differential Geom.
- Volume 104, Number 1 (2016), 59-109.

### On the anti-canonical geometry of $\mathbb{Q}$-Fano threefolds

Meng Chen and Chen Jiang

#### Abstract

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 $m \geq 97$.

#### Article information

**Source**

J. Differential Geom., Volume 104, Number 1 (2016), 59-109.

**Dates**

First available in Project Euclid: 6 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1473186539

**Digital Object Identifier**

doi:10.4310/jdg/1473186539

**Mathematical Reviews number (MathSciNet)**

MR3544286

**Zentralblatt MATH identifier**

1375.14137

#### Citation

Chen, Meng; Jiang, Chen. On the anti-canonical geometry of $\mathbb{Q}$-Fano threefolds. J. Differential Geom. 104 (2016), no. 1, 59--109. doi:10.4310/jdg/1473186539. https://projecteuclid.org/euclid.jdg/1473186539