Geometry of the Moduli of Higher Spin Curves: Irreducibility and Picard Group

This article treats various aspects of the geometry of the moduli Sgr of r-spin curves and its compactification Sgr. Generalized spin curves, or r-spin curves, are pairs (X, L) with X a smooth curve and L a line bundle whose r tensor power is isomorphic to the canonical bundle of X. These are a natural generalization of 2-spin curves (algebraic curves with a thetacharacteristic), which have been of interest lately because they are the subject of a remarkable conjecture of E. Witten. We show that the compactified moduli space Sgr is projective, that when r is odd, Sgr is irreducible, and that when r is even, Sgr is the disjoint union of two irreducible components. Furthermore, we generalize results of Cornalba describing and giving relations between many of the elements of the Picard group of Sgr and Sgr . And we show that when 2 or 3 divides r, then PicSgr has non-zero torsion.