Manuscripta Mathematica

, Volume 142, Issue 3–4, pp 409–437 | Cite as

The orbifold cohomology of moduli of genus 3 curves

  • N. Pagani
  • O. TommasiEmail author


In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of \({\mathcal{M}3}\) .

Mathematics Subject Classifications (2010)

Primary: 14H10 55N32 Secondary: 14N35 14D23 14H37 32G15 55P50 


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  1. 1.
    Abramovich D., Corti A., Vistoli A.: Twisted bundles and admissible covers. Special issue in honor of Steven L. Kleiman. Commun. Algebra 31(8), 3547–3618 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramovich, D., Graber, T., Vistoli, A.: Algebraic orbifold quantum products. In: Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., vol. 310, pp. 1–24. American Mathematical Society, Providence (2002)Google Scholar
  3. 3.
    Abramovich D., Graber T., Vistoli A.: Gromov–Witten theory of Deligne–Mumford stacks. Am. J. Math. 130(5), 1337–1398 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Abramovich D., Vistoli A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15(1), 27–75 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bayer A., Cadman C.: Quantum cohomology of [C Nr]. Compos. Math. 146(5), 1291–1322 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernstein, M.: Moduli of curves with level structure. Ph.D. Thesis, Harvard University (1999)Google Scholar
  7. 7.
    Cadman C.: Using stacks to impose tangency conditions on curves. Am. J. Math. 129(2), 405–427 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Catanese, F.: Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of \({Sing (\bar{\mathfrak M_g})}\) . arXiv:1011.0316v1Google Scholar
  9. 9.
    Chen W., Ruan Y.: A new cohomology theory of orbifold. Commun. Math. Phys. 248(1), 1–31 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cornalba M.: On the locus of curves with automorphisms. Ann. Mat. Pura Appl. 149(4), 135–151 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dolgachev, I.V.: Classical Algebraic Geometry. A Modern View, vol. xii. Cambridge University Press, Cambridge (2012)Google Scholar
  12. 12.
    Edmonds A.L.: Surface symmetry. I. Mich. Math. J. 29(2), 171–183 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fantechi, B.: Private communication (2008)Google Scholar
  14. 14.
    Getzler, E.: Operads and moduli of genus 0 Riemann surfaces. In: The Moduli Space of Curves (Texel Island, 1994), Progr. Math., vol. 129, pp. 199–230. Birkhäuser Boston, Boston (1995)Google Scholar
  15. 15.
    Getzler, E.: Topological recursion relations in genus 2. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), vol. 73–106. World Scientific Publishing, River Edge (1998)Google Scholar
  16. 16.
    Grothendieck A.: Sur quelques points d’algèbre homologique (French). Tôhoku Math. J. 9(2), 119–221 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kisin M., Lehrer G.I.: Equivariant Poincaré polynomials and counting points over finite fields. J. Algebra 247(2), 435–451 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Knudsen F.F.: Projectivity of the moduli space of stable curves II. The stacks \({\overline{M}_{g,n}}\) . Math. Scand. 52, 161–199 (1983)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Krug, S.: Rational cohomology of \({\bar R_2}\) (and \({\bar S_2)}\) . arXiv:1012.5191v1Google Scholar
  20. 20.
    Looijenga, E.; Cohomology of \({\mathcal{M}_3}\) and \({{\mathcal {M}^1_3}}\) . In: Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math., vol. 150, pp. 205–228. American Mathematical Society, Providence (1993)Google Scholar
  21. 21.
    Pagani, N.: Chen–Ruan cohomology of \({\mathcal{M}_{1,n}}\) and \({\overline{\mathcal{M}}_{1,n}}\) . To appear in Ann. Inst. Fourier (Grenoble) 63 (2013)Google Scholar
  22. 22.
    Pagani N.: The Chen–Ruan cohomology of moduli of curves of genus 2 with marked points. Adv. Math. 229(3), 1643–1687 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pagani, N.: The orbifold cohomology of moduli of hyperelliptic curves. Int. Math. Res. Not. 2012(10):2163–2178 (2012)Google Scholar
  24. 24.
    Pardini R.: Abelian covers of algebraic varieties. J. Reine Angew. Math. 417, 191–213 (1991)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Peters, C.A.M., Steenbrink, J.H.M.: Mixed Hodge Structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 52. Springer, Berlin (2008)Google Scholar
  26. 26.
    Spencer, J.E.: The orbifold cohomology of the moduli of genus-two curves. In: Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, 167–184. American Mathematical Society, Providence (2006)Google Scholar
  27. 27.
    Tommasi O.: Rational cohomology of the moduli space of genus 4 curves. Compos. Math. 141(2), 359–384 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tommasi O.: Rational cohomology of M 3,2. Compos. Math. 143(4), 986–1002 (2007)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany

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