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Torsion-free sheaves and moduli of generalized spin curves. (English) Zbl 0912.14010
In this article the author constructs and describes three compactifications of the space of generalized spin curves. Generalized spin curves, or $$r$$-spin curves, are pairs $$(X,L)$$ with $$X$$ a smooth curve, and $$L$$ a line bundle whose $$r$$-th tensor power is isomorphic to the canonical cotangent bundle of $$X.$$ These are a natural generalization of $$2$$-spin curves (algebraic curves with a theta-characteristic). The main results about quasi-spin curves are summed up in the following three theorems.
Theorem (algebraicity). Quasi-spin curves form a separated algebraic stack of finite type over $$\overline{M_g},$$ the moduli space of stable curves.
Theorem (density). The stack of smooth spin curves is dense in the stack of quasi-spin curves.
Theorem (properness). The stack of quasi-spin curves is proper over the stack of stable curves.
Three different compactifications over $$\mathbb Z\left[ \frac 1r\right] ,$$ all using torsion-free sheaves, are constructed. All three yield algebraic stack, one of which is shown to have Gorenstein singularities that can be described explicitly, and one of which is smooth. All three compactifications generalize constructions of Deligne and Cornalba done for the case when $$r=2$$.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14K25 Theta functions and abelian varieties 14M30 Supervarieties
##### Keywords:
spin curves; compactification; theta-characteristics; stack
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