Calabi-Yau pairs of complexity two
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908504870092800 |
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| author | Moraga, Joaquín Yáñez, José Ignacio |
| author_facet | Moraga, Joaquín Yáñez, José Ignacio |
| contents | A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface $X$ is cluster type if and only if $X$ has only $A$-type singularities and either $\mathrm{vol}(X)>1$ or $|X^{\rm sing}|\leq 3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Calabi-Yau pairs of complexity two Moraga, Joaquín Yáñez, José Ignacio Algebraic Geometry Primary: 14E30, 14D10, Secondary: 14M25, 14J17 A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface $X$ is cluster type if and only if $X$ has only $A$-type singularities and either $\mathrm{vol}(X)>1$ or $|X^{\rm sing}|\leq 3$. |
| title | Calabi-Yau pairs of complexity two |
| topic | Algebraic Geometry Primary: 14E30, 14D10, Secondary: 14M25, 14J17 |
| url | https://arxiv.org/abs/2412.18830 |