A compactification of the moduli space of multiple-spin curves.

Using line bundles on quasi-stable curves, we construct a smooth Deligne-Mumford compactification for the moduli space of curves with an m-tuple of spin structures. This compactification normalizes the m-fold product of Cornalba's compactification of curves with a single spin structure over the moduli space of stable curves. For all m, we give a combinatorial description of the local structure of the corresponding coarse moduli spaces and classify all irreducible components.