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

TL;DR

This work extends the bounds on the anticanonical volume for weak $\mathbb{Q}$-Fano threefolds with Picard rank two, proving that $-K_X^3$ is at most $64$ or equals $72$ with a unique geometric realization as a projective bundle over $\mathbb{P}^2$. The authors employ two-ray game techniques and a careful analysis of $K_X$-negative extremal contractions to classify potential geometries and fiber structures, yielding sharp upper bounds and identifying the exceptional case. The results build on and complement prior bounds, clarifying the extremal volumes and contributing to the broader classification program for weak $\mathbb{Q}$-Fano threefolds. Overall, the paper tightens volume bounds, isolates the sole higher-volume geometry, and enhances understanding of how extremal contractions control anticanonical volumes in this setting.

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 $ρ(X)\leq 2$, 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.

Figures (1)

• Figure 1: The last flip.

Theorems & Definitions (13)

• Theorem 2: =Theorem \ref{['thm:main']}
• Lemma 3: L*Lemma 5.1
• Lemma 4: L*Lemma 5.2
• Lemma 5
• proof
• Lemma 6
• proof
• Proposition 7
• proof
• Theorem 8