On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two

Abstract

We show that for a weak $\mathbb{Q}$-Fano threefold $X$ of Picard rank two ($\mathbb{Q}$-factorial with at worst terminal singularities), the anticanonical volume satisfies $-K_X^3\le 72$ except in one case, and the equality holds only if $X=\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^2}\oplus {\mathcal{O}}_{{\mathbb{P}}^2}(3))$. The approach in this article can serve as a general strategy to establish the optimal upper bound of $-K_X^3$ for any canonical Fano threefolds, where the described main result serves as the first step.

18 pages. Comments are welcome