Inventiones mathematicae

, Volume 194, Issue 1, pp 73–118 | Cite as

Syzygies of torsion bundles and the geometry of the level modular variety over \(\overline{\mathcal{M}}_{g}\)

  • Alessandro Chiodo
  • David Eisenbud
  • Gavril Farkas
  • Frank-Olaf Schreyer


We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of  in its Jacobian. These statements can be viewed an analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space \(\mathcal{R}_{g,\ell}\) of twisted level curves of genus g and use this to derive results about the birational geometry of \(\mathcal{R}_{g, \ell}\). For instance, we prove that \(\mathcal{R}_{g,3}\) is a variety of general type when g>11 and the Kodaira dimension of \(\mathcal{R}_{11,3}\) is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.


  1. 1.
    Abramovich, D., Corti, A., Vistoli, A.: Twisted bundles and admissible coverings. Commun. Algebra 31, 3547–3618 (2003) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aprodu, M., Nagel, J.: Koszul Cohomology and Algebraic Geometry. University Lecture Series, vol. 52. American Mathematical Society, Providence (2010) MATHGoogle Scholar
  3. 3.
    Abramovich, D., Vistoli, A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15, 27–75 (2002) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bauer, I., Catanese, F.: The rationality of certain moduli spaces of curves of genus 3. In: Cohomological and Geometric Approaches to Rationality Problems. Progress in Math., vol. 282. Birkhäuser, Boston (2010) Google Scholar
  5. 5.
    Bauer, I., Verra, A.: The rationality of the moduli space of genus-4 curves endowed with an order-3 subgroup of their Jacobian. Mich. Math. J. 59, 483–504 (2010) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Beauville, A.: Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions thêta. Bull. Soc. Math. Fr. 116, 431–448 (1988) MathSciNetMATHGoogle Scholar
  7. 7.
    Caporaso, L., Casagrande, C., Cornalba, M.: Moduli of roots of line bundles on curves. Trans. Am. Math. Soc. 359, 3733–3768 (2007) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chiodo, A.: Stable twisted curves and their r-spin structures. Ann. Inst. Fourier 58, 1635–1689 (2008) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chiodo, A.: Towards an enumerative geometry of the moduli space of twisted curves and rth roots. Compos. Math. 144, 1461–1496 (2008) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chiodo, A., Farkas, G.: Singularities of the moduli space of level curves. arXiv:1205.0201
  11. 11.
    Dumas, J.-G., Giorgi, P., Pernet, C.: Dense linear algebra over finite fields. ACM Trans. Math. Softw. 35(3) (2009). 34 pp. Google Scholar
  12. 12.
    Eisenbud, D.: Commutative Algebra. Graduate Texts in Mathematics, vol. 150. Springer, Berlin (2004) MATHGoogle Scholar
  13. 13.
    Eisenbud, D., Harris, J.: The Kodaira dimension of the moduli space of curves of genus ≥23. Invent. Math. 90, 359–387 (1987) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173–204 (1989) MathSciNetMATHGoogle Scholar
  15. 15.
    Eusen, F., Schreyer, F.-O.: A remark on a conjecture of Paranjape and Ramanan. In: Faber, C., Farkas, G., de Jong, R. (eds.) Geometry and Arithmetic, Schiermonnikoog 2010. EMS Series of Congress Reports, pp. 113–123 (2012) CrossRefGoogle Scholar
  16. 16.
    Farkas, G.: Koszul divisors on moduli spaces of curves. Am. J. Math. 131, 819–869 (2009) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Farkas, G., Ludwig, K.: The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. 12, 755–795 (2010) MathSciNetMATHGoogle Scholar
  18. 18.
    Farkas, G., Mustaţă, M., Popa, M.: Divisors on \(\mathcal{M}_{g, g+1}\) and the minimal resolution conjecture for points on canonical curves. Ann. Sci. Éc. Norm. Super. 36, 553–581 (2003) MATHGoogle Scholar
  19. 19.
    Farkas, G., Popa, M.: Effective divisors on \(\overline{\mathcal{M}}_{g}\), curves on K3 surfaces and the slope conjecture. J. Algebr. Geom. 14, 151–174 (2005) MathSciNetGoogle Scholar
  20. 20.
    Farkas, G., Verra, A.: The classification of the universal Jacobian over the moduli space of curves. Comment. Math. Helv. arXiv:1005.5354
  21. 21.
    Farkas, G., Verra, A.: Moduli of theta characteristics via Nikulin surfaces. Math. Ann. 354, 465–496 (2012) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Farkas, G., Verra, A.: \(\mathcal{R}_{8}\) is uniruled. Preprint Google Scholar
  23. 23.
    Green, M.: Koszul cohomology and the cohomology of projective varieties. J. Differ. Geom. 19, 125–171 (1984) MATHGoogle Scholar
  24. 24.
    Harris, J., Mumford, D.: On the Kodaira dimension of \(\overline{\mathcal{M}}_{g}\). Invent. Math. 67, 23–88 (1982) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jarvis, T.: Geometry of the moduli of higher spin curves. Int. J. Math. 11, 637–663 (2000) MathSciNetMATHGoogle Scholar
  26. 26.
    Jarvis, T.: The Picard group of the moduli of higher spin curves. N.Y. J. Math. 7, 23–47 (2001) MathSciNetMATHGoogle Scholar
  27. 27.
    Mukai, S.: Curves and Grassmannians. In: Yang, J.-H., Namikawa, Y., Ueno, K. (eds.) Algebraic Geometry and Related Topics, Inchon 1992, pp. 19–40. International Press, Somerville (1992) Google Scholar
  28. 28.
    Schreyer, F.-O.: A standard basis approach to syzygies of canonical curves. J. Reine Angew. Math. 421, 83–123 (1991) MathSciNetMATHGoogle Scholar
  29. 29.
    Graf v. Bothmer, H.-C.: Generic syzygy schemes. J. Pure Appl. Algebra 208, 867–876 (2007) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Verra, A.: The unirationality of the moduli space of curves of genus 14 or lower. Compos. Math. 141, 1425–1444 (2005) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Voisin, C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Compos. Math. 141, 1163–1190 (2005) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alessandro Chiodo
    • 1
  • David Eisenbud
    • 2
  • Gavril Farkas
    • 3
  • Frank-Olaf Schreyer
    • 4
  1. 1.Institut FourierUniversité de Grenoble, U.M.R. CNRS 5582Saint Martin d’HèresFrance
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany
  4. 4.Universität des SaarlandesSaarbrückenGermany

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