Stable twisted curves and their r-spin structures
Annales de l'Institut Fourier, Volume 58 (2008) no. 5, p. 1635-1689

The subject of this article is the notion of r-spin structure: a line bundle whose rth 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.

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 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 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 M g   ; chacune correspond à un multi-indice l =(l 0 ,l 1 ,) identifiant une notion de stabilité  : la l -stabilité. On détermine par la suite tout choix convenable de l pour lequel les structures r-spin forment un torseur fini au-dessus du champ des courbes l -stables.

DOI : https://doi.org/10.5802/aif.2394
Classification:  14H10,  14H60
Keywords: Spin structures, twisted curves, moduli of curves
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     author = {Chiodo, Alessandro},
     title = {Stable twisted curves and their $r$-spin structures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     pages = {1635-1689},
     doi = {10.5802/aif.2394},
     mrnumber = {2445829},
     zbl = {1179.14028},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_5_1635_0}
}
Chiodo, Alessandro. Stable twisted curves and their $r$-spin structures. Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1635-1689. doi : 10.5802/aif.2394. http://www.numdam.org/item/AIF_2008__58_5_1635_0/

[1] Abramovich, Dan Lectures on Gromov-Witten invariants of orbifolds (Preprint http://arxiv.org/abs/math//0512372)

[2] Abramovich, Dan; Corti, Alessio; Vistoli, Angelo Twisted bundles and admissible covers, Comm. Algebra, Tome 31 (2003) no. 8, pp. 3547-3618 (Special issue in honor of Steven L. Kleiman) | Article | MR 2007376 | Zbl 1077.14034

[3] Abramovich, Dan; Graber, Tom; Vistoli, Angelo Algebraic orbifold quantum products, Orbifolds in mathematics and physics (Madison, WI, 2001), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 310 (2002), pp. 1-24 | MR 1950940 | Zbl 1067.14055

[4] Abramovich, Dan; Jarvis, Tyler J. Moduli of twisted spin curves, Proc. Amer. Math. Soc., Tome 131 (2003) no. 3, p. 685-699 (electronic) | Article | MR 1937405 | Zbl 1037.14008

[5] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Amer. Math. Soc., Tome 15 (2002) no. 1, pp. 27-75 | Article | MR 1862797 | Zbl 0991.14007

[6] Arbarello, Enrico; Cornalba, Maurizio The Picard groups of the moduli spaces of curves, Topology, Tome 26 (1987) no. 2, pp. 153-171 | Article | MR 895568 | Zbl 0625.14014

[7] Artin, M. Versal deformations and algebraic stacks, Invent. Math., Tome 27 (1974), pp. 165-189 | Article | MR 399094 | Zbl 0317.14001

[8] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 21 (1990) | MR 1045822 | Zbl 0705.14001

[9] Breen, Lawrence Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 86 (1990), pp. 401-476 | MR 1086889 | Zbl 0743.14034

[10] Bryan, Jim; Graber, Tom The Crepant Resolution Conjecture (Preprint http://arxiv.org/abs/math/0610129) | Zbl 1198.14053

[11] Cadman, Charles Using stacks to impose tangency conditions on curves, Amer. J. Math., Tome 129 (2007) no. 2, pp. 405-427 | Article | MR 2306040 | Zbl 1127.14002

[12] Caporaso, Lucia; Casagrande, Cinzia; Cornalba, Maurizio Moduli of roots of line bundles on curves, Trans. Amer. Math. Soc., Tome 359 (2007) no. 8, p. 3733-3768 (electronic) | Article | MR 2302513 | Zbl 1140.14022 | Zbl pre05148115

[13] Chiodo, Alessandro Towards an enumerative geometry of the moduli space of twisted curves and r-th roots (Preprint: http://arxiv.org/abs/math/0607324) | MR 2238814 | Zbl 1166.14018

[14] Chiodo, Alessandro The Witten top Chern class via K-theory, J. Algebraic Geom., Tome 15 (2006) no. 4, pp. 681-707 | Article | MR 2237266 | Zbl 1117.14008

[15] Coates, Tom; Corti, Alessio; Iritani, Hiroshi; Tseng, Hsian-Hua Computing Genus-Zero Twisted Gromov-Witten Invariants (Preprint: http://arxiv.org/abs/math/0702234) | Zbl 1176.14009

[16] Cornalba, Maurizio Moduli of curves and theta-characteristics, Lectures on Riemann surfaces (Trieste, 1987), World Sci. Publ., Teaneck, NJ (1989), pp. 560-589 | MR 1082361 | Zbl 0800.14011

[17] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 75-109 | Article | Numdam | MR 262240 | Zbl 0181.48803

[18] Faber, C.; Shadrin, Sergey; Zvonkine, Dimitri Tautological relations and the r-spin Witten conjecture (Preprint: math.AG/0612510) | Zbl 1203.53090

[19] Grothendieck, Alexander Technique de descente et théorèmes d’existence en géométrie algébrique. II. Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki, Vol. 5, Soc. Math. France, Paris (1995), pp. Exp. No. 195, 369-390 | Numdam | Zbl 0234.14007

[20] Harer, John The second homology group of the mapping class group of an orientable surface, Invent. Math., Tome 72 (1983) no. 2, pp. 221-239 | Article | MR 700769 | Zbl 0533.57003

[21] Jarvis, T. J. Torsion-free sheaves and moduli of generalized spin curves, Compositio Math., Tome 110 (1998) no. 3, pp. 291-333 | Article | MR 1602060 | Zbl 0912.14010

[22] Jarvis, Tyler J. Geometry of the moduli of higher spin curves, Internat. J. Math., Tome 11 (2000) no. 5, pp. 637-663 | Article | MR 1780734 | Zbl 1094.14504

[23] Jarvis, Tyler J. The Picard group of the moduli of higher spin curves, New York J. Math., Tome 7 (2001), p. 23-47 (electronic) | MR 1838471 | Zbl 0977.14010

[24] Jarvis, Tyler J.; Kimura, Takashi; Vaintrob, Arkady Tensor products of Frobenius manifolds and moduli spaces of higher spin curves, Conférence Moshé Flato 1999, Vol. II (Dijon), Kluwer Acad. Publ., Dordrecht (Math. Phys. Stud.) Tome 22 (2000), pp. 145-166 | MR 1805911 | Zbl 0988.81120

[25] Kato, Kazuya Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD (1989), pp. 191-224 | MR 1463703 | Zbl 0776.14004

[26] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. of Math. (2), Tome 145 (1997) no. 1, pp. 193-213 | Article | MR 1432041 | Zbl 0881.14018

[27] Kontsevich, Maxim Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Tome 147 (1992) no. 1, pp. 1-23 | Article | MR 1171758 | Zbl 0756.35081

[28] Kresch, Andrew Cycle groups for Artin stacks, Invent. Math., Tome 138 (1999) no. 3, pp. 495-536 | Article | MR 1719823 | Zbl 0938.14003

[29] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Tome 39 (2000) | MR 1771927 | Zbl 0945.14005

[30] Lieblich, Max Remarks on the stack of coherent algebras, Int. Math. Res. Not. (2006), pp. Art. ID 75273, 12 | Article | MR 2233719 | Zbl 1108.14003

[31] Matsuki, Kenji; Olsson, Martin Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett., Tome 12 (2005) no. 2-3, pp. 207-217 | MR 2150877 | Zbl 1080.14023

[32] Mestrano, Nicole Conjecture de Franchetta forte, Invent. Math., Tome 87 (1987) no. 2, pp. 365-376 | Article | MR 870734 | Zbl 0585.14011

[33] Milne, James S. Étale cohomology, Princeton University Press, Princeton, N.J., Princeton Mathematical Series, Tome 33 (1980) | MR 559531 | Zbl 0433.14012

[34] Mumford, David Abelian varieties, Published for the Tata Institute of Fundamental Research, Bombay, Tata Institute of Fundamental Research Studies in Mathematics, No. 5 (1970) | MR 282985 | Zbl 0223.14022

[35] Olsson, Martin C. (Log) twisted curves, Compos. Math., Tome 143 (2007) no. 2, pp. 476-494 | MR 2309994 | Zbl 1138.14017 | Zbl pre05150517

[36] Polishchuk, Alexander; Vaintrob, Arkady Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 276 (2001), pp. 229-249 | Zbl 1051.14007

[37] Raynaud, Michel Spécialisation du foncteur de Picard. Critère numérique de représentabilité, C. R. Acad. Sci. Paris Sér. A-B, Tome 264 (1967), p. A1001-A1004 | MR 237515 | Zbl 0148.41702

[38] Romagny, Matthieu Sur quelques aspects des champs de revêtements de courbes algébriques, Institut Fourier,Université Grenoble I (2002) (Ph. D. Thesis)

[39] Witten, Edward Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA (1991), pp. 243-310 | MR 1144529 | Zbl 0757.53049

[40] Witten, Edward Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX (1993), pp. 235-269 | MR 1215968 | Zbl 0812.14017