Geometry of the Moduli of Higher Spin Curves

Abstract

Abstract. This article treats various aspects of the geometry of the moduli S 1/r g of r-spin curves and its compactification S 1/r g . 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 (KdVr) heirarchy. There is also a W -algebra conjecture for these spaces [16] generalizing the 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 S 1/r g,n (K) of triples (X, L, b) of a smooth curve X, a line bundle L on X, and an isomorphism b : L ⊗r -K. In the special case that K = ω is the relative dualizing sheaf, then S 1/r g,n (K) is the stack S 1/r g,n of r-spin curves.We construct a smooth compactification S 1/r g,n (K) of the stack S 1/r g,n (K), describe the geometric meaning of its points, and prove that it is projective.We also prove that when r is odd and g > 1, the compactified stack of spin curves S 1/r g and its coarse moduli space S 1/r g are irreducible, and when r is even and g > 1, S 1/r g 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 1/r 1 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 S 1/r g [15], and also in the study of the cohomological field theory related to Witten's conjecture [16,34].

Moduli of twisted spin curves

“…See [5] for a detailed proof. When k is congruent to k mod r, the two stacks M License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use…”

“…Many delicate features of r-spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the∂-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.The moduli space of smooth r-spin curves was compactified by the second author, using torsion free sheaves and coherent nets of torsion free sheaves in [4] and [5]. In order to construct a satisfactory compactification it was necessary in those papers to study in detail the behavior of torsion free sheaves with an r-power map.…”

“…The moduli space of smooth r-spin curves was compactified by the second author, using torsion free sheaves and coherent nets of torsion free sheaves in [4] and [5]. In order to construct a satisfactory compactification it was necessary in those papers to study in detail the behavior of torsion free sheaves with an r-power map.…”

The Witten top Chern class via 𝐾-theory

“…Remark (Witten top Chern class using other compactifications). In this paper we work with Jarvis's compactification [Jar00]. In fact, one can give a different compactification of the stack of smooth r-spin curves S g,n (r, k k k) by means of (balanced) twisted curves in the sense of Abramovich and Vistoli.…”

“…Definition of the moduli stack. The moduli stack of stable r-spin curves was first defined by Jarvis [Jar00] and is the compactification of the stack S g,n (r, k k k) labeled by the integer and nonnegative indexes r, g, n, and k k k = (k 1 , k 2 , . .…”