Tropical Derivation of Cohomology Ring of Heavy/Light Hassett Spaces

Abstract

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as \calm_{g, w} for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of \calm_{0, w}, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of \overline{\calm}_{0,w} as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology