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Alessandro Chiodo
Stable twisted curves and their $r$-spin structures
(Courbes champêtres stables et leurs structures $r$-spin)
Annales de l'institut Fourier, 58 no. 5 (2008), p. 1635-1689, doi: 10.5802/aif.2394
Article PDF | Reviews MR 2445829 | Zbl 1179.14028 | 1 citation in Cedram
Class. Math.: 14H10, 14H60
Keywords: Spin structures, twisted curves, moduli of curves

Résumé - Abstract

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 $\vec{l}=(l_0,l_1,\dots )$ identifying a notion of stability: $\vec{l}$-stability. Then, we determine the choices of $\vec{l}$ for which $r$-spin structures form a finite torsor over the moduli of $\vec{l}$-stable curves.

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