Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908352371490816 |
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| author | Jiang, Chen |
| author_facet | Jiang, Chen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03832 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II Jiang, Chen Algebraic Geometry 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. |
| title | Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2207.03832 |