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. 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 \(\bar{M}_{0, w}\).