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

Abstract

By a canonical (resp. terminal) weak $\mathbb{Q}$-Fano $3$-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $\mathbb{Q}$-Cartier, nef and big. For a canonical weak $\mathbb{Q}$-Fano $3$-fold $V$, we show that there exists a terminal weak $\mathbb{Q}$-Fano $3$-fold $X$, being birational to $V$, such that the $m$-th anti-canonical map defined by $|-m{K}_X|$ is birational for all $m\ge 52$. As an intermediate result, we show that for any $K$-Mori fiber space $Y$ of a canonical weak $\mathbb{Q}$-Fano $3$-fold, the $m$-th anti-canonical map defined by $|-m{K}_Y|$ is birational for all $m\ge 52$.

63 pages, comments are welcome; v2: final version, to appear in Annales de l'Institut Fourier

References in corpus (1)

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

Cited by in corpus (3)

• 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, III
• The Noether inequality for algebraic threefolds (With an Appendix by János Kollár)