Multiscale differentials and wonderful models

Abstract

We study the relationships between several varieties parametrizing marked curves with differentials in the literature. More precisely, we prove that the space ${\mathcal{B}}_n$ of multiscale differentials of genus 0 with $n+1$ marked points of orders $(0,\dots ,0,-2)$ is a wonderful variety. This shows that the Chow ring of ${\mathcal{B}}_n$ is generated by the classes of a collection of smooth boundary divisors with normal crossings subject to simple and explicit linear and quadratic relations. Furthermore, we realize ${\mathcal{B}}_n$ as a subvariety of the space ${\mathcal{A}}_n$ of multiscale lines and prove that ${\mathcal{B}}_n$ can be realized as the normalized Chow quotient of ${\mathcal{A}}_n$ by a natural ${\mathbb{C}}^{\ast }$-action.

20 pages, 3 figures