A Supplement to the anticanonical Volumes of weak $\mathbb{Q}$-Fano threefolds of Picard rank two

TL;DR

This paper refines the classification of weak $Q$-Fano threefolds with Picard rank at most two by establishing bounds on their anticanonical volumes, identifying a special case with explicit structure.

Contribution

It provides a supplementary result to prior work, precisely characterizing the anticanonical volume bounds and the unique case for Picard rank two threefolds.

Findings

If $ ho(X) leq 2$, then $-K_X^3 leq 64$ or equals 72 for a specific bundle.

The case $-K_X^3=72$ corresponds to a projective bundle over $P^2$.

The result narrows the possible volumes of such threefolds, aiding classification efforts.

Abstract

We show that for a weak $\mathbb{Q}$-Fano threefold $X$ ($\mathbb{Q}$-factorial with terminal singularities and $-K_X$ is nef and big) of Picard rank $\rho(X)\leq 2$, either $-K_X^3\leq 64$ or $-K_X^3=72$ and $X=\mathbb{P}_{\mathbb{P}^2}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. This is supplementary to the previous work in arXiv:2501.12555.