Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II

TL;DR

This paper characterizes certain terminal Fano threefolds with minimal anti-canonical volume as specific weighted hypersurfaces, providing explicit classifications for these extremal cases and related examples.

Contribution

It identifies the weighted hypersurface structure of terminal Fano 3-folds with minimal volume and characterizes similar examples in Iano-Fletcher's list.

Findings

Terminal Fano 3-fold with volume 1/330 is a weighted hypersurface in P(1,5,6,22,33).

Characterization of 11 examples of weighted hypersurfaces matching numerical data.

Extension of classification to other extremal Fano 3-folds in the same family.

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 $\mathbb{Q}$-factorial terminal Fano $3$-fold $X$ with $\rho(X)=1$ and $(-K_X)^3=\frac{1}{330}$ is a weighted hypersurface of degree $66$ in $\mathbb{P}(1,5,6,22,33)$. By the same method, we also give characterizations for other $11$ examples of weighted hypersurfaces of the form $X_{6d}\subset \mathbb{P}(1,a,b,2d,3d)$ in Iano-Fletcher's list. Namely, we show that if a $\mathbb{Q}$-factorial terminal Fano $3$-fold $X$ with $\rho(X)=1$ has the same numerical data as $X_{6d}$, then $X$ itself is a weighted hypersurface of the same type.