Multiscale differentials and wonderful models

TL;DR

The paper establishes a precise bridge between moduli spaces of genus-0 marked curves with prescribed differentials and the theory of wonderful models. It proves that the genus-0 multiscale differential moduli space $\mathcal{B}_n$ is the wonderful model $Y_n$, yielding a concrete, boundary-driven description of its Chow ring via linear and quadratic relations tied to the graphic matroid $M(K_n)$. It then connects $\mathcal{B}_n$ to the space $\mathcal{A}_n$ of multiscale lines by showing $\mathcal{B}_n$ sits as a boundary divisor in $\mathcal{A}_n$ and is the fixed locus of a natural $\mathbb{C}^*$-action; further, $\mathcal{B}_n$ is both the normalized Chow quotient and a GIT quotient of $\mathcal{A}_n$ under this action. The results unify combinatorial structures (partitions, rooted level trees, matroid Chow rings) with moduli of differentials, offering a robust framework for extending these correspondences to other differential data types.

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 $(0,\ldots,0,-2)$ 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 $\mathbb{C}^*$-action.

Figures (3)

• Figure 1: A comb curve in $\overline{M}_{0,10}$ corresponding to the partition $\pi = 124|58|36|7|9$ in $L_9$.
• Figure 2: Dual tree to boundary divisor in $\cA_n$ isomorphic to ${\mathcal{B}}_n$
• Figure 3: The fixed locus of $\mu$ for $n=3$ is $D \cup \{q,q_{12},q_{13},q_{23}\}$.

Theorems & Definitions (36)

• Theorem 1.1: \ref{['multiscale_wonderful']}
• Proposition 1.2
• Theorem 1.3: \ref{['thm: normalized Chow quotient']}
• Theorem 1.4: \ref{['thm: GIT quotient']}
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof