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

TL;DR

This paper proves that for terminal weak -Fano threefolds, the anti-canonical map becomes birational when m is at least 59, advancing understanding of their geometric properties.

Contribution

It establishes a uniform bound (m  59) ensuring the anti-canonical map is birational for all such threefolds, improving previous results.

Findings

The m-th anti-canonical map is birational for all m  59.

Provides a concrete bound for the birationality of anti-canonical maps.

Enhances understanding of the geometry of weak -Fano threefolds.

Abstract

For a terminal weak $\mathbb{Q}$-Fano $3$-fold $X$, we show that the $m$-th anti-canonical map defined by $|-mK_X|$ is birational for all $m\geq59$.