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

TL;DR

This paper establishes an upper bound of 72 for the anticanonical volume of weak Q-Fano threefolds with Picard rank two, identifying the unique case of equality and proposing a strategy for broader classes.

Contribution

It provides the first explicit upper bound for anticanonical volumes in this class of threefolds and introduces a general approach applicable to canonical Fano threefolds.

Findings

Anticanonical volume K_X^3;72 for weak Qano threefolds of Picard rank two.

Equality case occurs only for a specific projective bundle over P^2.

Proposes a strategy to determine optimal bounds for broader classes of Fano threefolds.

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 $-K_X^3\leq72$ 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.