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The Picard group of the moduli of higher spin curves. (English) Zbl 0977.14010
Summary: 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, describing and 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 by T. J. Jarvis, T. Kimura and A. Vaintrob [Compos. Math. 126, No. 2, 157-212 (2001; Zbl 1015.14028)]. We use these relations to show that when $$2$$ or $$3$$ divides $$r$$, the Picard group has non-zero torsion. And finally, we work out some specific examples.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C22 Picard groups 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 14H81 Relationships between algebraic curves and physics
##### Keywords:
Witten conjecture; Picard group; $$r$$-spin curves
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