On moduli of pointed real curves of genus zero

TL;DR

This paper introduces and stratifies the moduli space of pointed real genus zero curves, computes its first Stiefel-Whitney class, and constructs its orientation double cover, advancing understanding of its topological structure.

Contribution

It defines the moduli space of pointed real genus zero curves, describes its stratification via dual trees, and constructs the orientation double cover, providing new topological insights.

Findings

Calculated the first Stiefel-Whitney class of the moduli space.

Constructed the orientation double cover of the moduli space.

Provided a stratification of the moduli space via dual trees.

Abstract

We introduce the moduli space $R \bar{M}_{2k,l}$ of pointed real curves of genus zero and give its natural stratification. The strata of $R \bar{M}_{2k,l}$ correspond to real curves of genus zero with different degeneration types and are encoded by their dual trees with certain decorations. By using this stratification, we calculate the first Stiefel-Whitney class of $R \bar{M}_{2k,l}$ and construct the orientation double cover $R \widetilde{M}_{2k,l}$$ of $R \bar{M}_{2k,l}$.