Blowdowns of the Deligne-Mumford Spaces of Real Rational Curves

We describe a sequence of smooth quotients of the Deligne-Mumford moduli space $\mathbb{R}{\bar{\mathcal{M}}}_{0,\mathrm{\ell }+1}$ of real rational curves with $\mathrm{\ell }\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}1$ conjugate pairs of marked points that terminates at $\mathbb{R}{\bar{\mathcal{M}}}_{0,\mathrm{\ell }}\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbb{C}{\mathbb{P}}^1$. This produces an analogue of Keel’s blowup construction of the Deligne-Mumford moduli spaces ${\bar{\mathcal{M}}}_{\mathrm{\ell }+1}$ of rational curves with $\mathrm{\ell }\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}1$ 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 $\mathbb{R}{\bar{\mathcal{M}}}_{0,\mathrm{\ell }+1}$ to completely determine the rational (co)homology ring of $\mathbb{R}{\bar{\mathcal{M}}}_{0,\mathrm{\ell }}$.