The Cohomology Ring of the Deligne-Mumford Moduli Space of Real Rational Curves with Conjugate Marked Points

Abstract

It is a long-established and heavily-used fact that the integral cohomology ring of the Deligne-Mumford moduli space of (complex) rational curves is the polynomial ring on the boundary divisors modulo the ideal generated by the obvious geometric relations between them. We show that the rational cohomology ring of the Deligne-Mumford moduli space of real rational curves with conjugate marked points only is the polynomial ring on certain (``complex") boundary divisors and real boundary hypersurfaces modulo the ideal generated by the obvious geometric relations between them and the geometric relation in positive dimension and codimension identified in a previous paper.

Figures (5)

• Figure 1: The basic cases of (\ref{['cM04rel_e2']}), equivalences of three points in $H^2(\overline{\mathcal{M}}_4;\mathbb Z)$.
• Figure 2: The first line represents basic cases of (\ref{['RcM04rel_e2a']}) and (\ref{['RcM04rel_e2b']}), an equivalence of two points in $H^1(\mathbb R\overline{\mathcal{M}}_{0,2};\mathbb Q)$ and a relation between three loops in $H^2(\mathbb R\overline{\mathcal{M}}_{0,3};\mathbb Q)$. The bottom diagram represents the intersection pattern of the six boundary divisors $\mathbb R D_{I;J,K}\!\approx\!S^1$ with the four boundary hypersurfaces $\mathbb R E_{J',K'}\!\approx\!S^2$ in $\mathbb R\overline{\mathcal{M}}_{0,3}$. The former are labeled by the unique element of $I\!\subset\![3]$ and the sign of $(-1)^{|J|}\!=\!(-1)^{|K|}$; the latter are labeled by the subset $J',K'\!\subset\![3]$ not containing 1.
• Figure 3: A basic case of (\ref{['RcM04rel_e2b2']}), a relation between four loops in $H^2(\mathbb R\overline{\mathcal{M}}_{0,3};\mathbb Q)$.
• Figure 4: Generic representatives of the top boundary strata $D_{\varrho}$ 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 5: The structure and orientation of the Deligne-Mumford compactification $\overline{\mathcal{M}}_{0,2}^{\tau}\!\subset\!\mathbb R\overline{\mathcal{M}}_{0,2}$ of $\mathcal{M}_{0,2}^{\tau}$.

Theorems & Definitions (37)

• Theorem 2.1: Keel
• Theorem 2.2
• Remark 2.3
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• proof : Proof of Proposition \ref{['BlHom_prp']}
• Lemma 3.4
• proof
• Proposition 4.1