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 $\rho (X)\le 2$, either $-K_X^3\le 64$ 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.

13 pages, 1 figure. Comments are welcome