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

TL;DR

This paper studies the geometry of $Q$-Fano 3-folds, proving that their anti-$m$-canonical maps are birational for all $m$ beyond certain bounds, with specific results for weak $Q$-Fano 3-folds.

Contribution

It establishes explicit bounds on $m$ for the birationality of anti-$m$-canonical maps for $Q$-Fano 3-folds and weak $Q$-Fano 3-folds, advancing understanding of their anti-canonical geometry.

Findings

Anti-$m$-canonical map is birational for all $m \,\geq 39$ on $Q$-Fano 3-folds.

For weak $Q$-Fano 3-folds, the map is birational for all $m \,\geq 97$.

Provides geometric insights into the structure of $Q$-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 $|-mK_X|$ in this paper 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 $m\geq 97$.