Moduli spaces of witch curves topologically realize the 2-associahedra

TL;DR

This paper constructs a compactified moduli space of witch curves, stratifies it by 2-associahedra, and shows its topological properties, linking it to the moduli space of stable disk trees.

Contribution

It introduces a new topological realization of 2-associahedra via moduli spaces of witch curves, establishing their compactness and stratification.

Findings

The moduli space $ar{2 ext{M}}_{ extbf{n}}$ is compact and metrizable.

The stratification of $ar{2 ext{M}}_{ extbf{n}}$ matches the 2-associahedron $W_{ extbf{n}}$.

The forgetful map to $ar{ ext{M}}_r$ is continuous and stratification-preserving.

Abstract

For $r \geq 1$ and $\mathbf{n} \in \mathbb{Z}_{\geq0}^r\setminus\{\mathbf{0}\}$, we construct the compactified moduli space $\overline{2\mathcal{M}}_{\mathbf{n}}$ of witch curves of type $\mathbf{n}$. We equip $\overline{2\mathcal{M}}_{\mathbf{n}}$ with a stratification by the 2-associahedron $W_{\mathbf{n}}$, and prove that $\overline{2\mathcal{M}}_{\mathbf{n}}$ is compact and metrizable. In addition, we show that the forgetful map $\overline{2\mathcal{M}}_{\mathbf{n}} \to \overline{\mathcal{M}}_r$ to the moduli space of stable disk trees is continuous and respects the stratifications.