The Picard Group of the Moduli of Higher Spin Curves

TL;DR

This paper investigates the structure of the Picard group of the moduli space of r-spin curves, revealing relations and torsion properties, and contributes to understanding Witten's conjecture in algebraic geometry.

Contribution

It generalizes previous results on the Picard group of r-spin moduli stacks, establishing new relations and torsion properties relevant to Witten's conjecture.

Findings

Relations between elements of the Picard group are established.

When 2 or 3 divides r, the Picard group has non-zero torsion.

Specific examples for small genus and r are worked out.

Abstract

This article treats the Picard group of the moduli (stack) of r-spin curves and its compactification. Generalized spin curves, or r-spin curves are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because they are the subject of a remarkable conjecture of E. Witten, and because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. We generalize results of Cornalba, giving relations between many of the elements of the Picard group of the stacks. These relations are important in the proof of the genus-zero case of Witten's conjecture given in math.AG/9905034. We use these relations to show that when 2 or 3 divides r, then the Picard group of the open stack has non-zero torsion. And finally, we work out some specific examples for small values of g and r.