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

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.