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

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 $m \geq 97$.