Stable twisted curves and their $r$-spin structures

Chiodo, Alessandro

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1635-1689.

L’objet de cet article est la notion de structure $r$-spin  : un fibré en droites dont la puissance $r$-ième est isomorphe au fibré canonique. Au-dessus du champ ${\mathsf{M}}_g$ des courbes lisses de genre $g$, les structures $r$-spin forment un torseur fini sous le groupe des fibrés de $r$-torsion. Au-dessus du champ ${\overline{\mathsf{M}}}_g$ des courbes stables de genre $g$, les structures $r$-spin forment un champ étale, mais la finitude et la structure de torseur ne sont pas préservées.

On améliore drastiquement cet état de choses si on resitue le problème dans la catégorie des courbes champêtres (“twisted curves” au sens d’Abramovich et Vistoli). On trouve d’abord que, dans cette catégorie, il existe plusieurs compactifications de ${\mathsf{M}}_g$  ; chacune correspond à un multi-indice $\overrightarrow{l}=(l_0,l_1,\cdots )$ identifiant une notion de stabilité  : la $\overrightarrow{l}$-stabilité. On détermine par la suite tout choix convenable de $\overrightarrow{l}$ pour lequel les structures $r$-spin forment un torseur fini au-dessus du champ des courbes $\overrightarrow{l}$-stables.

The subject of this article is the notion of $r$-spin structure: a line bundle whose $r$th power is isomorphic to the canonical bundle. Over the moduli functor ${\mathsf{M}}_g$ of smooth genus-$g$ curves, $r$-spin structures form a finite torsor under the group of $r$-torsion line bundles. Over the moduli functor ${\overline{\mathsf{M}}}_g$ of stable curves, $r$-spin structures form an étale stack, but both the finiteness and the torsor structure are lost.

In the present work, we show how this bad picture can be definitely improved just by placing the problem in the category of Abramovich and Vistoli’s twisted curves. First, we find that within such a category there exist several different compactifications of ${\mathsf{M}}_g$; each one corresponds to a different multiindex $\overrightarrow{l}=(l_0,l_1,\cdots )$ identifying a notion of stability: $\overrightarrow{l}$-stability. Then, we determine the choices of $\overrightarrow{l}$ for which $r$-spin structures form a finite torsor over the moduli of $\overrightarrow{l}$-stable curves.

Classification : 14H10, 14H60
Keywords: Spin structures, twisted curves, moduli of curves
Mot clés : structures $r$-spin, courbes champêtres, modules de courbes

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