Chow rings of stacks of prestable curves II

TL;DR

This paper studies the Chow ring of the moduli stack of prestable curves, showing in genus 0 it matches the tautological ring and providing a complete description with generators and relations, extending previous results.

Contribution

It provides a complete description of the Chow ring of ,n, generalizing earlier work and introducing new methods using boundary stratification and higher Chow groups.

Findings

Chow ring of ,n equals the tautological ring in genus 0.

Complete generators and relations for the Chow ring in genus 0.

New proof of results by Kontsevich and Manin.

Abstract

We continue the study of the Chow ring of the moduli stack $\mathfrak{M}_{g,n}$ of prestable curves begun in [arXiv:2012.09887v2]. In genus $0$, we show that the Chow ring of $\mathfrak{M}_{0,n}$ coincides with the tautological ring and give a complete description in terms of (additive) generators and relations. This generalizes earlier results by Keel and Kontsevich-Manin for the spaces of stable curves. Our argument uses the boundary stratification of the moduli stack together with the study of the first higher Chow groups of the strata, in particular providing a new proof of the results of Kontsevich and Manin.