On the geometry of the moduli space of spin curves

Ludwig, Katharina

doi:10.1090/S1056-3911-09-00505-0

Author: Katharina Ludwig
Journal: J. Algebraic Geom. 19 (2010), 133-171
DOI: https://doi.org/10.1090/S1056-3911-09-00505-0
Published electronically: April 23, 2009
MathSciNet review: 2551759
Full-text PDF

Abstract | References | Additional Information

Abstract: We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification $\overline {S}_g$ of the moduli space $S_g$ of spin curves, i.e., smooth curves of genus $g$ with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of $\overline {S}_g$ extends holomorphically to a desingularisation of $\overline {S}_g$.

References

Dan Abramovich and Tyler J. Jarvis, Moduli of twisted spin curves, Proc. Amer. Math. Soc. 131 (2003), no. 3, 685–699. MR 1937405, DOI https://doi.org/10.1090/S0002-9939-02-06562-0
Michael F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. MR 286136
Gilberto Bini and Claudio Fontanari, Moduli of curves and spin structures via algebraic geometry, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3207–3217. MR 2216264, DOI https://doi.org/10.1090/S0002-9947-06-03838-4
Lucia Caporaso and Cinzia Casagrande, Combinatorial properties of stable spin curves, Comm. Algebra 31 (2003), no. 8, 3653–3672. Special issue in honor of Steven L. Kleiman. MR 2007379, DOI https://doi.org/10.1081/AGB-120022437
Lucia Caporaso, Cinzia Casagrande, and Maurizio Cornalba, Moduli of roots of line bundles on curves, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3733–3768. MR 2302513, DOI https://doi.org/10.1090/S0002-9947-07-04087-1

Alessandro Chiodo. Higher spin curves and Witten’s top Chern class. PhD thesis, University of Cambridge, 2003.

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

Maurizio Cornalba, A remark on the Picard group of spin moduli space, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2 (1991), no. 3, 211–217 (English, with Italian summary). MR 1135424

P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240

David Eisenbud and Joe Harris, The Kodaira dimension of the moduli space of curves of genus $\geq 23$, Invent. Math. 90 (1987), no. 2, 359–387. MR 910206, DOI https://doi.org/10.1007/BF01388710

Gavril Farkas. $M_{22}$ is of general type. www.ma.utexas.edu/\symbol{126}gfarkas/.

Carel Faber, Sergey Shadrin, and Dimitri Zvonkine. Tautological relations and the $r$-spin Witten conjecture. arXiv:math/0612510 [math.AG], Dec 2006.

Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88. With an appendix by William Fulton. MR 664324, DOI https://doi.org/10.1007/BF01393371

Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825

T. J. Jarvis, Torsion-free sheaves and moduli of generalized spin curves, Compositio Math. 110 (1998), no. 3, 291–333. MR 1602060, DOI https://doi.org/10.1023/A%3A1000209527158

Tyler J. Jarvis, Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000), no. 5, 637–663. MR 1780734, DOI https://doi.org/10.1142/S0129167X00000325

S. Kallel and D. Sjerve. Invariant spin structures on Riemann surfaces. preprint, 2006.

David Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375–386. MR 210944

Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 273–310. MR 605348

Yung-Sheng Tai, On the Kodaira dimension of the moduli space of abelian varieties, Invent. Math. 68 (1982), no. 3, 425–439. MR 669424, DOI https://doi.org/10.1007/BF01389411

Additional Information

Katharina Ludwig
Affiliation: Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
Email: ludwig@math.uni-hannover.de

Received by editor(s): August 2, 2007
Received by editor(s) in revised form: October 18, 2007
Published electronically: April 23, 2009