Geometry of the moduli of higher spin curves

This article treats various aspects of the geometry of the moduli S-g(1/r) of r-spin curves and its compactification (S) over bar(g)(1/r). 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 of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand- Dikii (KdV(r)) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle K on the universal curve over the stack of stable curves, there is a smooth stack O-g,n(1/r)(K) of triples (X, L, b) of a smooth curve X, a line bundle L on X, and an isomorphism b : L-xr --> K. In the special case that K = omega is the relative dualizing sheaf, then O-g,n(1/r)(K) is the stack O-g,n(1/r) of r-spin curves. We construct a smooth compactification (O) over bar(g,n)(1/r) (K) of the stack O-g,n(1/r) (K), describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g > 1, the compactified stack of spin curves (O) over bar(g)(1/r) and its coarse moduli space (S) over bar(g)(1/r) are irreducible) and when r is even and g > 1, (S) over bar(g)(1/r) is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when g = 1, (S) over bar(1)(1/r) is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of (O) over bar(g)(1/r) [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].