Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings

Veronica Arena; Terry Dekun Song

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arXiv · 2605.17922

Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings

Veronica Arena·Terry Dekun Song

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Abstract

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.

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BibTeX RIS

Veronica Arena, Terry Dekun Song. 2026-05-18. Logarithmic Hilbert schemes of curves as weighted blow-ups and their integral Chow rings. https://arxiv.org/abs/2605.17922

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