Moduli spaces of witch curves topologically realize the 2-associahedra

Abstract

For $r\ge 1$ and $\mathbf{n}\in {\mathbb{Z}}_{\ge 0}^r\setminus \{0\}$, 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.

arXiv comment21 pages, 9 figures. Final version for publication in Journal of Symplectic Geometry