Chow Rings of Heavy/Light Hassett Spaces via Tropical Geometry

TL;DR

This paper computes the Chow ring of heavy/light Hassett spaces, revealing their structure via tropical geometry and relating it to known moduli space Chow rings, with explicit relations and generators.

Contribution

It provides a new presentation of the Chow ring for heavy/light Hassett spaces using tropical geometry and extends Keel's work to these weighted moduli spaces.

Findings

Chow ring of $ar{M}_{0,w}$ is computed explicitly.

Relation ideal for the Chow ring is characterized.

Pullback under Hassett's morphism identifies the Chow ring with a subring of $ar{M}_{0,n}$.

Abstract

We compute the Chow ring of an arbitrary heavy/light Hassett space $\bar{M}_{0, w}$. These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector $w$ consists of only heavy and light weights. Work of Cavalieri et al. exhibits these spaces as tropical compactifications of hyperplane arrangement complements. The computation of the Chow ring then reduces to intersection theory on the toric variety of the Bergman fan of a graphic matroid. Keel has calculated the Chow ring $A^*(\bar{M}_{0, n})$ of the moduli space $\bar{M}_{0, n}$ of stable nodal $n$-marked rational curves; his presentation is in terms of divisor classes of stable trees of $\mathbb{P}^1$'s having one nodal singularity. Our presentation of the ideal of relations for the Chow ring $A^*(\bar{M}_{0, w})$ is analogous. We show that pulling back under Hassett's birational reduction morphism $\rho_w: \bar{M}_{0, n} \to \bar{M}_{0, w}$ identifies the Chow ring $A^*(\bar{M}_{0, w})$ with the subring of $A^*(\bar{M}_{0, n})$ generated by divisors of $w$-stable trees, which are those trees which remain stable in $\bar{M}_{0, w}$.