An optimal boundedness on weak $\bQ$-Fano threefolds

TL;DR

This paper establishes new bounds on the anti-canonical volume and plurigenera of terminal weak Q-Fano threefolds, providing optimal universal lower bounds and specific positivity results.

Contribution

It proves that certain plurigenera are positive or greater than one and establishes an optimal universal lower bound for the anti-canonical volume of these threefolds.

Findings

P_{-6}(X)>0 for terminal weak Q-Fano threefolds

P_{-8}(X)>1 for these threefolds

Anti-canonical volume has a universal lower bound of 1/330

Abstract

Let $X$ be a terminal weak $\bQ$-Fano threefold. We prove that $P_{-6}(X)>0$ and $P_{-8}(X)>1$. We also prove that the anti-canonical volume has a universal lower bound $-K_X^3 \geq 1/330$. This lower bound is optimal.