Weak Fano threefolds arising as the blowup of a hyperquadric in $\mathbb{P}^4$ along a curve
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908827241152512 |
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| author | Schnattinger, Anne |
| author_facet | Schnattinger, Anne |
| contents | We characterize smooth irreducible curves $C$ on a smooth hyperquadric $Y$ of $\mathbb{P}^4$ such that the blowup of $Y$ along $C$ is a weak Fano threefold. These are precisely the smooth irreducible curves $C$ of degree $d$ and genus $g$ lying on a smooth hypercubic section of $Y$ such that (i) $C$ has no 4-secant line and no 7-secant conic; (ii) $d< 18$ and $(g,d)\not \in \{(4,7),\:(10, 11)\}$; (iii) either $3d-26<g\leq\frac{d^2-1}{12}$ or $(g,d)\in \{(4,6),\:(13,12)\}$. We prove the geometric realizability of each case, thereby proving the existence of weak Fano threefolds and Sarkisov links constructed from them, which were previously known only as numerical possibilities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak Fano threefolds arising as the blowup of a hyperquadric in $\mathbb{P}^4$ along a curve Schnattinger, Anne Algebraic Geometry 14E05, 14E30, 14J28, 14J45, 14J30, 14H99 We characterize smooth irreducible curves $C$ on a smooth hyperquadric $Y$ of $\mathbb{P}^4$ such that the blowup of $Y$ along $C$ is a weak Fano threefold. These are precisely the smooth irreducible curves $C$ of degree $d$ and genus $g$ lying on a smooth hypercubic section of $Y$ such that (i) $C$ has no 4-secant line and no 7-secant conic; (ii) $d< 18$ and $(g,d)\not \in \{(4,7),\:(10, 11)\}$; (iii) either $3d-26<g\leq\frac{d^2-1}{12}$ or $(g,d)\in \{(4,6),\:(13,12)\}$. We prove the geometric realizability of each case, thereby proving the existence of weak Fano threefolds and Sarkisov links constructed from them, which were previously known only as numerical possibilities. |
| title | Weak Fano threefolds arising as the blowup of a hyperquadric in $\mathbb{P}^4$ along a curve |
| topic | Algebraic Geometry 14E05, 14E30, 14J28, 14J45, 14J30, 14H99 |
| url | https://arxiv.org/abs/2512.20315 |