Moduli spaces of witch curves topologically realize the $2$-associahedra

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

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Received 5 December 2017

Accepted 9 May 2018

Published 17 January 2020