On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909463724687360 |
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| author | Lai, Ching-Jui |
| author_facet | Lai, Ching-Jui |
| contents | We show that for a weak $\mathbb{Q}$-Fano threefold $X$ of Picard rank two ($\mathbb{Q}$-factorial with at worst terminal singularities), the anticanonical volume satisfies $-K_X^3\leq72$ except in one case, and the equality holds only if $X=\mathbb{P} (\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. The approach in this article can serve as a general strategy to establish the optimal upper bound of $-K_X^3$ for any canonical Fano threefolds, where the described main result serves as the first step. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two Lai, Ching-Jui Algebraic Geometry 14J30, 14J45, 14E30 We show that for a weak $\mathbb{Q}$-Fano threefold $X$ of Picard rank two ($\mathbb{Q}$-factorial with at worst terminal singularities), the anticanonical volume satisfies $-K_X^3\leq72$ except in one case, and the equality holds only if $X=\mathbb{P} (\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(3))$. The approach in this article can serve as a general strategy to establish the optimal upper bound of $-K_X^3$ for any canonical Fano threefolds, where the described main result serves as the first step. |
| title | On anticanonical volumes of weak $\mathbb{Q}$-Fano terminal threefolds of Picard rank two |
| topic | Algebraic Geometry 14J30, 14J45, 14E30 |
| url | https://arxiv.org/abs/2501.12555 |