Moduli of twisted spin curves

Abstract

In this note we give a new, natural construction of a compactification of the stack of smooth r r -spin curves, which we call the stack of stable twisted r r -spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible G m \mathbb G_{\mathbf {m}} -spaces and Q \mathbb Q -line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves. We construct representable morphisms from the stacks of stable twisted r r -spin curves to the stacks of stable r r -spin curves and show that they are isomorphisms. Many delicate features of r r -spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the ∂ ¯ \bar \partial -operator of Seeley and Singer and Witten’s cohomology class go through without complications in the setting of twisted spin curves.