Moduli of twisted spin curves

In this note we give a new, natural construction of a compactification of the stack of smooth $r$-spin curves, which we call the stack of stable twisted $r$-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 $\mathbb G_{\mathbf{m}}$-spaces and $\mathbb Q$-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 $r$-spin curves to the stacks of stable $r$-spin curves and show that they are isomorphisms. 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 $\bar\partial$-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.