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

TL;DR

This paper shows that the logarithmic Hilbert scheme of points on a smooth curve can be described as an iterated weighted blow-up of the symmetric product, enabling explicit computation of their Chow rings.

Contribution

It explicitly identifies the structure of logarithmic Hilbert schemes as weighted blow-ups and computes their Chow rings, connecting to recent advances in algebraic geometry.

Findings

Logarithmic Hilbert scheme of points is an iterated weighted blow-up.

Chow rings of these schemes are computed explicitly.

Logarithmic Hilbert scheme of toric P^1 is a toric stack.

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.