Blowdowns of the Deligne-Mumford Spaces of Real Rational Curves

TL;DR

This paper constructs a sequence of smooth quotients of real Deligne-Mumford moduli spaces of rational curves, providing explicit blowup descriptions and an approach to determine their rational cohomology rings.

Contribution

It introduces an explicit blowup construction for real moduli spaces of rational curves, extending Keel's approach to the real case with detailed intermediate spaces.

Findings

Explicit description of blowups for real moduli spaces

Construction of intermediate spaces in the blowup sequence

Framework for computing rational cohomology rings

Abstract

We describe a sequence of smooth quotients of the Deligne-Mumford moduli space ${\mathbb R}\overline{\mathcal M}_{0,\ell+1}$ of real rational curves with $\ell\!+\!1$ conjugate pairs of marked points that terminates at ${\mathbb R}\overline{\mathcal M}_{0,\ell}\!\times\!{\mathbb C}{\mathbb P}^1$. This produces an analogue of Keel's blowup construction of the Deligne-Mumford moduli spaces $\overline{\mathcal M}_{\ell+1}$ of rational curves with $\ell\!+\!1$ marked points, but with an explicit description of the intermediate spaces and the blowups of three different types. The same framework readily adapts to the real moduli spaces with real points. In a sequel, we use this inductive construction of ${\mathbb R}\overline{\mathcal M}_{0,\ell+1}$ to completely determine the rational (co)homology ring of ${\mathbb R}\overline{\mathcal M}_{0,\ell}$.