Torsion-free Sheaves and Moduli of Generalized Spin Curves

This article treats the compactification of the space of higher spin curves, i.e. pairs (X,L) with L an r root of the canonical bundle of X. More precisely, for positive integers r and g, with g > 2, r dividing 2g − 2, and for a flat family of smooth curves f : X → T, an r-spin structure on X is a line bundle L such that L ∼= ωX /T . And an r-spin curve over T is a flat family of smooth curves with an r-spin structure. Now, for a fixed base scheme S over Z[1/r], let Spinr,g be the sheafification of the functor which takes an S-scheme T to the set of isomorphism classes of r-spin curves over T. A compactification of the space of spin curves is a space (scheme or algebraic stack), which is proper over Mg (the Deligne-Mumford compactification of the space of curves), and whose fibre over Mg represents, at least coarsely, the functor Spinr,g. It is possible (see [18]) to compactify Spinr,g using geometric invariant theory. Namely, in the style of L. Caporaso [3], for a fixed d >> 0 one can choose a subscheme of the Hilbert scheme Hilb P with a geometric quotient that coarsely represents Spinr,g. And using results of Gieseker (c.f. [11, Theorems 1.0.0 and 1.0.1], ) one can show that the semi-stable closure of the subscheme in Hilb P has a categorical quotient that provides a compactification. This compactification is actually a subscheme of Caporaso’s compactification of the relative Picard scheme over Mg. The principle drawback to the GIT compactification is that it is not obviously the solution to a moduli problem, and therefore it is difficult to describe the resulting space and to make the construction work over a general base, rather than only over algebraically closed fields. Moreover, the GIT construction requires that one make some arbitrary choices, and it is not clear that the resulting compactification is completely independent of these choices. Therefore, the approach we take here is to pose a moduli problem, using torsion-free sheaves, and then show that the associated stack is actually algebraic and that it does indeed compactify Spinr,g. We discuss three different moduli problems that provide compactifications and describe some of their characteristics. The naive approach would be to use a rankone torsion-free sheaf E with a suitable OX -module homomorphism from E ⊗r to the canonical bundle. But this doesn’t quite work, as the resulting space is not separated. Some additional conditions on the cokernel of the homomorhism are