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 FarkasEmail author
  • 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.


Modulus Space Line Bundle Hilbert Series Rational Curf Kodaira Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are grateful to Anand Patel for pointing out an error in an earlier version of this paper, to Burcin Eröcal and Florian Geiss for help with the computational aspects, and to Hans-Christian Graf von Bothmer and Alessandro Verra, who made suggestions that allowed us to improve the results of Sect. 5.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alessandro Chiodo
    • 1
  • David Eisenbud
    • 2
  • Gavril Farkas
    • 3
    Email author
  • 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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