Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II
Chen Jiang
TL;DR
This work determines the structure of terminal Fano $3$-folds with the smallest anti-canonical volume by proving that any $\mathbb{Q}$-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 $\mathbb{P}(1,5,6,22,33)$ given by $F=t^2+F_0$, and extends this classification to 11 additional families from Iano–Fletcher's list, showing that sharing the same numerical data forces $X$ to be of the same hypersurface type. The approach combines Reid's Riemann–Roch formula with detailed analysis of anti-pluri-canonical systems, using generators $f,g,h,p,q$ of the anti-canonical ring to embed $X$ birationally into a weighted projective space and prove $X$ is isomorphic to a hypersurface $Y$ defined by $t^2+F_0=0$. An alternative pathway via Noether–Fano–Iskovkhikh inequalities and the general elephant conjecture is discussed, outlining how canonicity of the relevant pair could yield the same hypersurface realization. Overall, the paper tightens the link between minimal anti-canonical volume, birational geometry, and the Iano–Fletcher weighted-hypersurface catalog.
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 $ρ(X)=1$ and $(-K_X)^3=\frac{1}{330}$ is a weighted hypersurface of degree $66$ in $\mathbb{P}(1,5,6,22,33)$. 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 $ρ(X)=1$ has the same numerical data as $X_{6d}$, then $X$ itself is a weighted hypersurface of the same type.