The object of this paper is the notion of r-spin structure: a line bundle
whose r-th power is isomorphic to the canonical bundle. Over the moduli functor
M_g of smooth genus-g curves, r-spin structures form a finite torsor under
the group of r-torsion line bundles. Over the moduli functor Mbar_g of stable
curves, r-spin structures form an 'etale stack, but the finiteness and the
torsor structure are lost.
In the present work, we show how this bad picture can be definitely improved
simply by placing the problem in the category of Abramovich and Vistoli's
twisted curves. First, we find that within such category there exist several
different compactifications of M_g; each one corresponds to a different
multiindex \ell=(l0,l1,...) identifying a notion of stability: \ell-stability.
Then, we determine the suitable choices of \ell for which r-spin structures
form a finite torsor over the moduli of \ell-stable curves.Comment: 44 pages, revised version, to appear in Annales de l'Institut Fourie
