Characterizing terminal Fano threefolds with the smallest anti-canonical volume

TL;DR

This paper classifies certain minimal volume terminal Fano threefolds, showing that the non-rational case with minimal anti-canonical volume is a specific weighted hypersurface.

Contribution

It identifies the unique structure of non-rational, -factorial terminal Fano threefolds with minimal volume, completing the classification at the lower bound.

Findings

Minimal anti-canonical volume is /330.

Such threefolds are weighted hypersurfaces in (1,5,6,22,33).

The non-rational case is explicitly characterized.

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 $\rho(X)=1$ and $(-K_X)^3=\frac{1}{330}$ is a weighted hypersurface of degree $66$ in $\mathbb{P}(1,5,6,22,33)$.