Stable twisted curves and their r -spin structures

Alessandro Chiodo[1]

  • [1] Université de Grenoble I Institut Fourier UMR 5582 CNRS 100 rue des Maths BP 74 38402 St Martin d’H�res (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 5, page 1635-1689
  • ISSN: 0373-0956

Abstract

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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 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 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 M g ; each one corresponds to a different multiindex l → = ( l 0 , l 1 , ⋯ ) identifying a notion of stability: l → -stability. Then, we determine the choices of l → for which r -spin structures form a finite torsor over the moduli of l → -stable curves.

How to cite

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Chiodo, Alessandro. "Stable twisted curves and their $r$-spin structures." Annales de l’institut Fourier 58.5 (2008): 1635-1689. <http://eudml.org/doc/10358>.

@article{Chiodo2008,
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.},
affiliation = {Université de Grenoble I Institut Fourier UMR 5582 CNRS 100 rue des Maths BP 74 38402 St Martin d’H�res (France)},
author = {Chiodo, Alessandro},
journal = {Annales de l’institut Fourier},
keywords = {Spin structures; twisted curves; moduli of curves; moduli stack of curves; spin structures},
language = {eng},
number = {5},
pages = {1635-1689},
publisher = {Association des Annales de l’institut Fourier},
title = {Stable twisted curves and their $r$-spin structures},
url = {http://eudml.org/doc/10358},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Chiodo, Alessandro
TI - Stable twisted curves and their $r$-spin structures
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1635
EP - 1689
AB - 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.
LA - eng
KW - Spin structures; twisted curves; moduli of curves; moduli stack of curves; spin structures
UR - http://eudml.org/doc/10358
ER -

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