On the anti-canonical geometry of $\mathbb{Q}$-Fano 3-folds

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 $|-m{K}_X|$ in this paper and prove that the anti-$m$-canonical map ${\phi }_{-m}$ is birational onto its image for all $m\ge 39$. 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 ${\phi }_{-m}$ is birational onto its image for all $m\ge 97$.

45 pages, to appear in Journal of Differential Geometry

Cited by in corpus (6)

• Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors
• An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds
• On the anti-canonical geometry of weak $\mathbb{Q}$-Fano threefolds, II
• Characterizing terminal Fano threefolds with the smallest anti-canonical volume
• On the anti-canonical geometry of weak $\mathbb{Q}$-Fano threefolds, III
• Arithmetic and geometric deformations of 3-folds