Chow rings of stacks of prestable curves II

Abstract

Abstract We continue the study of the Chow ring of the moduli stack 𝔐 g , n {\mathfrak{M}_{g,n}} of prestable curves begun in [Y. Bae and J. Schmitt, Chow rings of stacks of prestable curves I, Forum Math. Sigma 10 2022, Paper No. e28]. In genus 0, we show that the Chow ring of 𝔐 0 , n {\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 by Kontsevich and 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.