Chow rings of moduli spaces of genus 0 curves with collisions

TL;DR

The paper advances our understanding of Chow rings for genus 0 moduli spaces with collisions by introducing and exploiting simplicially stable compactifications $\mathcal{M}\overline{\,}_{0,\mathcal{K}}$. It develops a boundary-driven presentation framework that uses higher Chow groups and the motivic Künneth property to reduce computations to boundary strata indexed by $\mathcal{K}$-stable graphs, and it proves an explicit presentation in terms of boundary divisors with WDVV-type and intersection relations. A key outcome is a short, self-contained proof that the Chow ring $\mathrm{CH}(\overline{\mathcal{M}}_{0,\mathcal{K}})$ is generated by boundary divisors with relations arising from codimension-1 intersections and WDVV relations pushed forward from $\mathrm{CH}(\overline{\mathcal{M}}_{0,S})$, and, in the discrete case, this recovers Keel's classical presentation. The results provide a unified, genus-0 framework for computing CH rings of a broad family of modular compactifications and highlight the power of MKP and higher Chow techniques in boundary stratifications. Altogether, the work extends Keel-type descriptions to a wider class of moduli spaces and offers a robust toolkit for future boundary-CH computations in moduli theory.

Abstract

Introduced in [BB], simplicially stable spaces are alternative compactifications of $\mathcal{M}_{g,n}$ generalizing Hassett's moduli spaces of weighted stable curves. We give presentations of the Chow rings of these spaces in genus $0$ using techniques developed by the author in [New25]. When considering the special case of $\overline{\mathcal{M}}_{0,n}$, this gives a new proof of Keel's presentation of $\operatorname{CH}(\overline{\mathcal{M}}_{0,n})$.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 2.5
• Proposition 2.6
• Remark 2.7
• Definition 3.1