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

An rr-Spin Riemann surface is a Riemann surface equipped with a choice of rrth root of the (co)tangent bundle. We give a careful construction of the moduli space (orbifold) of rr-Spin Riemann surfaces, and explain how to establish a Madsen–Weiss theorem for it. This allows us to prove the “Mumford conjecture” for these moduli spaces, but more interestingly allows us to compute their algebraic Picard groups (for g≥10g≥10, or g≥9g≥9 in the 2-Spin case). We give a complete description of these Picard groups, in terms of explicitly constructed line bundles.