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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2503.06612 |
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| _version_ | 1866912266805313536 |
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| author | Peng, Junyao |
| author_facet | Peng, Junyao |
| contents | We study special $\mathbb{G}_m$-equivariant degenerations of a smooth del Pezzo surface $X$ induced by valuations that are log canonical places of $(X,C)$ for a nodal anti-canonical curve $C$. We show that the space of special valuations in the dual complex of $(X,C)$ is connected and admits a locally finite partition into sub-intervals, each associated to a $\mathbb{G}_m$-equivariant degeneration of $X$. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Kollár valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathbb{G}_m$-Equivariant Degenerations of del Pezzo Surfaces Peng, Junyao Algebraic Geometry 14E99 We study special $\mathbb{G}_m$-equivariant degenerations of a smooth del Pezzo surface $X$ induced by valuations that are log canonical places of $(X,C)$ for a nodal anti-canonical curve $C$. We show that the space of special valuations in the dual complex of $(X,C)$ is connected and admits a locally finite partition into sub-intervals, each associated to a $\mathbb{G}_m$-equivariant degeneration of $X$. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Kollár valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement. |
| title | $\mathbb{G}_m$-Equivariant Degenerations of del Pezzo Surfaces |
| topic | Algebraic Geometry 14E99 |
| url | https://arxiv.org/abs/2503.06612 |