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.

[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, Volume 31 (2003) no. 8, pp. 3547-3618 (Special issue in honor of Steven L. Kleiman) | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

[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

[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 | Zbl

[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 | DOI | EuDML | Numdam | MR | Zbl

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

[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 | EuDML | Numdam | Zbl

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

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

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

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

[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) (Math. Phys. Stud.), Volume 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 145-166 | MR | Zbl

[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 | Zbl

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

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

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

[29] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, 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], 39, Springer-Verlag, Berlin, 2000 | MR | Zbl

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

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

[32] Mestrano, Nicole Conjecture de Franchetta forte, Invent. Math., Volume 87 (1987) no. 2, pp. 365-376 | DOI | EuDML | MR | Zbl

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

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

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

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

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

[38] Romagny, Matthieu Sur quelques aspects des champs de revêtements de courbes algébriques, Institut Fourier,Université Grenoble I (2002) (Ph. D. Thesis http://www-fourier.ujf-grenoble.fr/THESE/ps/t120.ps.gz)

[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 | Zbl

[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 | Zbl