The Picard group of the moduli space of r-Spin Riemann surfaces

TL;DR

This paper proves a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, enabling the computation of their stable rational cohomology and Picard groups, and provides a complete description of these groups for genus at least 9.

Contribution

It establishes a homological stability theorem for r-Spin Riemann surfaces and explicitly describes their Picard groups for high genus.

Findings

Complete description of the integral Picard groups for genus ≥ 9

Identification of generators for the Picard groups

Relations between generators are explicitly determined

Abstract

We have recently proved a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem for these moduli spaces. This allows us to effectively study their stable cohomology, and to compute their stable rational cohomology and their integral Picard groups. Using these methods we give a complete description of their integral Picard groups for genus at least 9 in terms of geometrically defined generators, and determine the relations between them.