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

TL;DR

This paper proves that for certain weak $Q$-Fano threefolds with canonical or terminal singularities, the anti-canonical map becomes birational for all sufficiently large multiples, specifically for all $m \, \geq \, 52$, establishing a uniform bound.

Contribution

It demonstrates the existence of a uniform bound (m ≥ 52) for the birationality of anti-canonical maps on weak $Q$-Fano threefolds with canonical or terminal singularities, and relates these to Mori fiber spaces.

Findings

Existence of a terminal weak $Q$-Fano threefold birational to a given canonical one with birational anti-canonical maps for m ≥ 52.

Anti-canonical maps are birational for all m ≥ 52 on Mori fiber spaces of canonical weak $Q$-Fano threefolds.

Provides a uniform bound for the birationality of anti-canonical maps in this class of threefolds.

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 $|-mK_{X}|$ is birational for all $m\geq 52$. 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 $|-mK_{Y}|$ is birational for all $m\geq 52$.