Wahl singularities in degenerations of del Pezzo surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912475427897344 |
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| author | Urzúa, Giancarlo Zúñiga, Juan Pablo |
| author_facet | Urzúa, Giancarlo Zúñiga, Juan Pablo |
| contents | For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities are classified using the Markov equation. To achieve this, we introduce del Pezzo Wahl chains with markings. They define marked del Pezzo surfaces $W_{*m}$ that govern all such degenerations. We also prove that every marked del Pezzo surface degenerates into a canonically defined toric del Pezzo surface with only T-singularities. In addition, we establish a one-to-one correspondence between the $W_{*m}$ surfaces and certain fake weighted projective planes. As applications, we show that every Wahl singularity occurs for del Pezzo surfaces of degree $\leq 4$, but that there are infinitely many Wahl singularities that do not arise for degrees $\geq 5$. We also use Hacking's exceptional collections to provide geometric proofs of recent results by Polishchuk and Rains on exceptional vector bundles on del Pezzo surfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wahl singularities in degenerations of del Pezzo surfaces Urzúa, Giancarlo Zúñiga, Juan Pablo Algebraic Geometry Geometric Topology Symplectic Geometry For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities are classified using the Markov equation. To achieve this, we introduce del Pezzo Wahl chains with markings. They define marked del Pezzo surfaces $W_{*m}$ that govern all such degenerations. We also prove that every marked del Pezzo surface degenerates into a canonically defined toric del Pezzo surface with only T-singularities. In addition, we establish a one-to-one correspondence between the $W_{*m}$ surfaces and certain fake weighted projective planes. As applications, we show that every Wahl singularity occurs for del Pezzo surfaces of degree $\leq 4$, but that there are infinitely many Wahl singularities that do not arise for degrees $\geq 5$. We also use Hacking's exceptional collections to provide geometric proofs of recent results by Polishchuk and Rains on exceptional vector bundles on del Pezzo surfaces. |
| title | Wahl singularities in degenerations of del Pezzo surfaces |
| topic | Algebraic Geometry Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2504.19929 |