Characterizing terminal Fano threefolds with the smallest anti-canonical volume

TL;DR

The paper shows that the smallest anti-canonical volume case for a non-rational, $\mathbb{Q}$-factorial terminal Fano 3-fold with $\rho(X)=1$, namely $(-K_X)^3=\frac{1}{330}$, is realized by a weighted hypersurface of degree $66$ in $\mathbb{P}(1,5,6,22,33)$, denoted $X_{66}$. It combines Reid's Riemann–Roch formula (fixing the Reid basket and Hilbert series) with a section-ring analysis to produce explicit generators $f_1,f_5,f_6,f_{22},f_{33}$ and a defining equation $F=t^2+F_0$ in the ambient weighted projective space, showing $X\simeq \operatorname{Proj} R(X,-K_X) \simeq Y$. The result affirms that such $X$ is exactly a quasi-smooth weighted hypersurface $X_{66}$ and highlights a method potentially extendable to other small-volume terminal Fano 3-folds. This contributes a concrete geometric model and supports the rigidity phenomena associated with $X_{66}$, offering a pathway to characterize other extremal cases in the Fano landscape.

Abstract

It was proved by J. A. Chen and M. Chen that a terminal Fano $3$-fold $X$ satisfies $(-K_X)^3\geq \frac{1}{330}$. We show that a non-rational $\mathbb{Q}$-factorial terminal Fano $3$-fold $X$ with $ρ(X)=1$ and $(-K_X)^3=\frac{1}{330}$ is a weighted hypersurface of degree $66$ in $\mathbb{P}(1,5,6,22,33)$.

Theorems & Definitions (10)

• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Remark 3.2
• proof : Proof of Theorem \ref{['mainthm']}
• Claim 3.3
• proof