Chow rings of heavy/light Hassett spaces via tropical geometry

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. [3] 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 [16] has calculated the Chow ring $A^{⁎}\left({\bar{M}}_{0,n}\right)$ 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 $P^1$'s having one nodal singularity. Our presentation of the ideal of relations for the Chow ring $A^{⁎}\left({\bar{M}}_{0,w}\right)$ 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 $A^{⁎}\left({\bar{M}}_{0,w}\right)$ with the subring of $A^{⁎}\left({\bar{M}}_{0,n}\right)$ generated by divisors of w-stable trees, which are those trees which remain stable in ${\bar{M}}_{0,w}$.