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

TL;DR

This paper determines the structure of the rational cohomology ring of the moduli space of real rational curves with conjugate marked points, showing it is generated by boundary divisors with relations similar to the complex case.

Contribution

It extends the understanding of the cohomology ring from complex to real rational curves, identifying generators and relations specific to the real case.

Findings

Cohomology ring is a polynomial ring on boundary divisors and hypersurfaces.

Relations include geometric relations known from complex case and new real-specific relations.

Provides a complete algebraic description of the cohomology ring for the real moduli space.

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.