Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916022493118464 |
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| author | Arena, Veronica Song, Terry Dekun |
| author_facet | Arena, Veronica Song, Terry Dekun |
| contents | The logarithmic Hilbert scheme of a logarithmic curve parametrizes subschemes on the expanded degenerations of the curve that are transverse to the boundary. We prove that the logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the underlying curve. In doing so, we explicitly identify the blow-up centers, weights, and give them modular interpretations. As applications, we calculate their integral Chow rings in terms of those of the symmetric products. Key ingredients in our work include two recent results: the integral Chow ring formula of weighted blow-ups and a weighted analogue of Castelnuovo's criterion for blow-downs. We recover the folklore result the logarithmic Hilbert scheme of toric $\mathbb{P}^1$ is a toric stack, and the Appendix by Dhruv Ranganathan outlines a complementary approach using Chow quotients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings Arena, Veronica Song, Terry Dekun Algebraic Geometry 14D23, 14C15 The logarithmic Hilbert scheme of a logarithmic curve parametrizes subschemes on the expanded degenerations of the curve that are transverse to the boundary. We prove that the logarithmic Hilbert scheme of points on a smooth pointed curve is an iterated weighted blow-up of the symmetric product of the underlying curve. In doing so, we explicitly identify the blow-up centers, weights, and give them modular interpretations. As applications, we calculate their integral Chow rings in terms of those of the symmetric products. Key ingredients in our work include two recent results: the integral Chow ring formula of weighted blow-ups and a weighted analogue of Castelnuovo's criterion for blow-downs. We recover the folklore result the logarithmic Hilbert scheme of toric $\mathbb{P}^1$ is a toric stack, and the Appendix by Dhruv Ranganathan outlines a complementary approach using Chow quotients. |
| title | Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings |
| topic | Algebraic Geometry 14D23, 14C15 |
| url | https://arxiv.org/abs/2605.17922 |