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

TL;DR

This work establishes an optimal upper bound of $-K_X^3$ for weak $\mathbb{Q}$-Fano threefolds with Picard number two and terminal singularities, showing $-K_X^3\le72$ with equality only for $X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$; the proof uses a two-ray game that analyzes the two extremal contractions of $X$, decomposes $-K_X$ into two nef classes, and treats separately the cases where both contractions yield Mori fiber spaces or where a $K$-trivial divisorial contraction occurs. In the first case the bound tightens to $-K_X^3\le54$, while in the second case one obtains $-K_X^3\le72$ with equality only for the stated projective bundle; these results rely on Reid's Riemann-Roch and the geometry of conic bundles. The paper also explicates how this strategy can extend to canonical Fano threefolds and discusses optimal examples, including a comparison to the canonical Fano $Y=\mathbb{P}(1,1,4,6)$, which attains the same bound but with a higher Picard rank upon terminalization. Overall, the work provides a sharp, case-distinguished framework for bounding anticanonical volumes in dimension three and offers a concrete optimal example within the $ ho=2$ weak $\mathbb{Q}$-Fano class.

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.

Theorems & Definitions (33)

• Definition 1.1
• Definition 1.2
• Theorem 1.4: CC
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Theorem 2.2
• Proposition 2.3
• proof