Blowdowns of the Deligne-Mumford Spaces of Real Rational Curves

TL;DR

This work develops a Keel‑style inductive blowup framework for the Deligne–Mumford spaces of real rational curves by constructing a chain of smooth quotients $X_{\varrho^*}$ starting from $\mathbb{R}\overline{\mathcal M}_{0,\ell+1}$ and terminating at $\mathbb{R}\overline{\mathcal M}_{0,\ell}\times\mathbb{C}\mathbb P^1$. It first recasts the complex case as explicit quotients of $\overline{\mathcal M}_{\ell+1}$ with holomorphic blowups along centers $Y_{\varrho^*-1;\varrho}^0$, using cross‑ratios to provide local coordinates and a non‑inductive construction. The real case is then treated, introducing three blowup types—real, complex, and augmented—whose centers intersect cleanly and whose exceptional loci reflect the top boundary strata of $\mathbb{R}\overline{\mathcal M}_{0,\ell}$, with orientability and chart‑dependent subtleties carefully handled. Together, these results lay the groundwork for a complete determination of the rational (co)homology of $\mathbb{R}\overline{\mathcal M}_{0,\ell}$ in a sequel, and they provide a concrete, explicit, and computable structure for the real moduli spaces via a sequence of controlled blowups.

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 ${\mathbb R}\overline{\mathcal M}_{0,\ell}$.

Figures (6)

• Figure 1: The three divisors $D_{4;\varrho}$ in $\overline{\mathcal{M}}_4\!\approx\!\mathbb C\mathbb P^1$.
• Figure 2: The three boundary hypersurfaces $D_{2;\varrho}$ in $\mathbb{R}\overline{\mathcal{M}}_{0,2}\!\approx\!\mathbb{R}\mathbb P^1$ and two of the six boundary divisors $D_{3;\varrho}$ in the orientable threefold $\mathbb{R}\overline{\mathcal{M}}_{0,3}$. Each line or circle represents $\mathbb C\mathbb P^1$. Each double-headed arrow labeled $\sigma$ indicates the involution on the corresponding real curve.
• Figure 3: Generic representatives of the top boundary strata $D_{\ell;\varrho}$ in $\mathbb{R}\overline{\mathcal{M}}_{0,\ell}$ and $\widetilde{D}_{\varrho}^{\bullet}$ in $\mathbb{R}\overline{\mathcal{M}}_{0,\ell+1}$. Each line or circle represents $\mathbb C\mathbb P^1$. Each double-headed arrow labeled $\sigma$ indicates the involution on the corresponding real curve.
• Figure 4: A stable 11-marked rational curve $\mathcal{C}$, its dual 11-marked tree $\Gamma$, the subtrees $(\textnormal{Ver}_{\widetilde{e}},\textnormal{Edg}_{\widetilde{e}})$ and $(\textnormal{Ver}_{\widetilde{e}}^c,\textnormal{Edg}_{\widetilde{e}}^c)$ of $\Gamma$ determined by the oriented edge $\widetilde{e}$ of $\Gamma$, and the subsets (\ref{['ellGavdfn_e']}) of [11] determined by a systematic marking map $\eta$.
• Figure 5: The three possibilities for $z_{\ell+1}(\widetilde{\mathcal{C}}_r)\!\in\!\mathbb C\mathbb P^1_v$ in the proof of Proposition \ref{['Cblowup_prp']} and in Remark \ref{['Cblowup_rmk']}

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 4.1
• proof