Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularites via key varieties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912714571382784 |
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| author | Takagi, Hiromichi |
| author_facet | Takagi, Hiromichi |
| contents | We consider the classification problem of prime $\mathbb{Q}$-Fano 3-folds with at most $1/2(1,1,1)$-singularities, which was initiated in [Taka2]. We construct two distinct classes of such 3-folds with genus one and six $1/2(1,1,1)$-singularities, each equipped with a prescribed Sarkisov link. Our method involves constructing certain higher-dimensional $\mathbb{Q}$-Fano varieties $Σ$, referred to as key varieties, by extending the Sarkisov links to higher dimensions. We prove that each such 3-fold $X$ arises as a linear section of the corresponding key variety $Σ$, and conversely, any general linear section of $Σ$ yields such an $X$. Various geometric properties of the key varieties $Σ$ are also investigated and clarified. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularites via key varieties Takagi, Hiromichi Algebraic Geometry 14J45, 14E30, 14N05 We consider the classification problem of prime $\mathbb{Q}$-Fano 3-folds with at most $1/2(1,1,1)$-singularities, which was initiated in [Taka2]. We construct two distinct classes of such 3-folds with genus one and six $1/2(1,1,1)$-singularities, each equipped with a prescribed Sarkisov link. Our method involves constructing certain higher-dimensional $\mathbb{Q}$-Fano varieties $Σ$, referred to as key varieties, by extending the Sarkisov links to higher dimensions. We prove that each such 3-fold $X$ arises as a linear section of the corresponding key variety $Σ$, and conversely, any general linear section of $Σ$ yields such an $X$. Various geometric properties of the key varieties $Σ$ are also investigated and clarified. |
| title | Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularites via key varieties |
| topic | Algebraic Geometry 14J45, 14E30, 14N05 |
| url | https://arxiv.org/abs/2507.13657 |