Skip to main content
SpringerLink
Log in

The Moduli Space of Curves and Gromov–Witten Theory

  • Chapter
Enumerative Invariants in Algebraic Geometry and String Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1947))

These notes are intended to explain how Gromov–Witten theory has been useful in understanding the moduli space of complex curves. We will focus on the moduli space of smooth curves and how much of the recent progress in understanding it has come through “enumerative” invariants in Gromov–Witten theory, something which we take for granted these days, but which should really be seen as surprising. There is one sense in which it should not be surprising — in many circumstances, modern arguments can be loosely interpreted as the fact that we can understand curves in general by studying branched covers of the complex projective line, as all curves can be so expressed. We will see this theme throughout the notes, from a Riemannstyle parameter count in Sect. 2.2 to the tool of relative virtual localization in Gromov—Witten theory in Sect. 5.

These notes culminate in an approach to Faber’s intersection number conjecture using relative Gromov–Witten theory (joint work with Goulden and Jackson [GJV3]). One motivation for this article is to convince the reader that our approach is natural and straightforward.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Arbarello and M. Cornalba, The Picard groups of the moduli spaces of curves, Topology 26 (1987), 153–171.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Arcara and Y.-P. Lee, Tautological equations in genus 2 via invariance conjectures, preprint 2005, math.AG/0502488.

    Google Scholar 

  3. D. Arcara and Y.-P. Lee, Tautological equations in \({\overline{M}}_{3,1}\) via invariance conjectures, preprint 2005, math.AG/0503184.

    Google Scholar 

  4. D. Arcara and F. Sato, Recursive formula for \({\psi }^{g} - l{a}_{1}{\psi }^{g-1} + \mathrel{\cdots } + {(-1)}^{g}{\lambda }_{g}\) in \({\overline{{\cal M}}}_{g,1}\), preprint 2006, math.AG/0605343.

    Google Scholar 

  5. V. I. Arnol’d, Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges, Funct. Anal. and its Appl. 30 no. 1 (1996), 1–17.

    Article  Google Scholar 

  6. M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Belorousski and R. Pandharipande, A descendent relation in genus 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 1, 171–191.

    MathSciNet  Google Scholar 

  8. K. Costello, Higher-genus Gromov–Witten invariants as genus 0 invariants of symmetric products, preprint 2003, math.AG/0303387.

    Google Scholar 

  9. D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical surveys and Monographs 68, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  10. M. Crescimanno and W. Taylor, Large N phases of chiral QCD 2, Nuclear Phys. B 437 (1995), 3–24.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Dénes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Ins. Hungar. Acad. Sci. 4 (1959), 63–70.

    MATH  Google Scholar 

  12. R. Dijkgraaf, H. Verlinde, and E. Verlinde, Topological strings in d < 1, Nuclear Phys. B 352 (1991), 59–86.

    Article  MathSciNet  Google Scholar 

  13. D. Edidin, Notes on the construction of the moduli space of curves, in Recent progress in intersection theory (Bologna, 1997), 85–113, Trends Math., Birkhäuser Boston, Boston, MA, 2000.

    Google Scholar 

  14. T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 1175–1180.

    MATH  MathSciNet  Google Scholar 

  15. T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), 297–327.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Faber, A conjectural description of the tautological ring of the moduli space of curves, in Moduli of Curves and Abelian Varieties, 109–129, Aspects Math., E33, Vieweg, Braunschweig, 1999.

    Google Scholar 

  17. C. Faber, Maple program for computing Hodge integrals, personal communication. Available at http://math.stanford.edu/ vakil/programs/.

  18. C. Faber, Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians, in New trends in algebraic geometry (Warwick, 1996), 93–109, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  19. C. Faber, personal communication, January 8, 2006.

    Google Scholar 

  20. C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. (Fulton volume) 48 (2000), 215–252.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the λ g conjecture, Ann. Math. 157 (2003), 97–124.

    Article  MATH  MathSciNet  Google Scholar 

  22. C. Faber and R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. 7 (2005), no. 1, 13–49.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Fantechi, Stacks for everybody, in European Congress of Mathematics, Vol. I (Barcelona, 2000), 349–359, Progr. Math., 201, Birkhäuser, Basel, 2001.

    Google Scholar 

  24. B. Fantechi and R. Pandharipande, Stable maps and branch divisors, Compositio Math. 130 (2002), no. 3, 345–364.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Gathmann, Absolute and relative Gromov–Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002), 171–203.

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Getzler, Intersection theory on \({\overline{{\cal M}}}_{1,4}\) and elliptic Gromov–Witten invariants, J. Amer. Math. Soc. 10 (1997), no. 4, 973–998.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Getzler, Topological recursion relations in genus 2, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 73–106, World Sci. Publishing, River Edge, NJ, 1998.

    Google Scholar 

  28. E. Getzler and R. Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B 530 (1998), 701–714.

    Article  MATH  MathSciNet  Google Scholar 

  29. J. Giansiracusa and D. Maulik, Topology and geometry of the moduli space of curves, on-line collection of resources on moduli of curves and related subjects, http://www.aimath.org/WWN/modspacecurves/.

  30. A. Givental, Gromov–Witten invariants and quantization of quadratic hamiltonians, Mosc. Math. J. 1 (2001), no. 4, 551–568, 645.

    MATH  MathSciNet  Google Scholar 

  31. I. P. Goulden and D. M. Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997), 51–60.

    Article  MATH  MathSciNet  Google Scholar 

  32. I. P. Goulden and D. M. Jackson, The number of ramified coverings of the sphere by the double torus, and a general form for higher genera, J. Combin. Theory A 88 (1999) 259–275.

    Article  MATH  MathSciNet  Google Scholar 

  33. I. P. Goulden, D. M. Jackson, R. Vakil, The Gromov–Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc. (3) 83 (2001), 563–581.

    Article  MathSciNet  Google Scholar 

  34. I. P. Goulden, D. M. Jackson, R. Vakil, Towards the geometry of double Hurwitz numbers, Adv. Math. (Artin issue) 198 (2005), 43–92.

    Article  MATH  MathSciNet  Google Scholar 

  35. I. P. Goulden, D. M. Jackson, R. Vakil, On Faber’s intersection number conjecture on the moduli space of curves, in preparation.

    Google Scholar 

  36. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), 487–518.

    Article  MATH  MathSciNet  Google Scholar 

  37. T. Graber and R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math. 135 (2003), no. 1, 25–36.

    Article  MATH  MathSciNet  Google Scholar 

  38. T. Graber and R. Vakil, On the tautological ring of \({\overline{{\cal M}}}_{g,n}\), in Proceedings of the Seventh Gökova Geometry-Topology Conference 2000, International Press, 2000.

    Google Scholar 

  39. T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005), no. 1, 1–37.

    Article  MATH  MathSciNet  Google Scholar 

  40. R. Hain and E. Looijenga, Mapping class groups and moduli spaces of curves, Proc. Sympos. Pure Math. 62 Part 2, pp. 97–142, Amer. Math. Soc., Providence, RI, 1997.

    MathSciNet  Google Scholar 

  41. J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), no. 3, 457–485.

    Article  MATH  MathSciNet  Google Scholar 

  42. J. Harris and I. Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag, New York, 1998.

    Google Scholar 

  43. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Gromov–Witten Theory and Mirror Symmetry, Clay Math. Inst., Amer. Math. Soc., 2002.

    Google Scholar 

  44. A. Hurwitz, Über Riemann’sche Flächen mit gegeben Verzweigungspunkten, Math. Ann. 39 (1891), 1–60.

    Article  MathSciNet  Google Scholar 

  45. E. Ionel, Topological recursive relations in H 2g(Mg,n), Invent. Math. 148 (2002), no. 3, 627–658.

    Article  MATH  MathSciNet  Google Scholar 

  46. E. Ionel, Relations in the tautological ring of M g, Duke Math. J. 129 (2005), no. 1, 157–186.

    Article  MATH  MathSciNet  Google Scholar 

  47. E.-N. Ionel and T. H. Parker, Relative Gromov–Witten invariants, Ann. of Math. (2) 157 (2003), 45–96.

    Article  MathSciNet  Google Scholar 

  48. E.-N. Ionel and T. H. Parker, The symplectic sum formula for Gromov–Witten invariants, Ann. of Math. (2), 159 (2004), 935–1025.

    Article  MathSciNet  Google Scholar 

  49. S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574.

    Article  MATH  MathSciNet  Google Scholar 

  50. Y.-S. Kim and K. Liu, A simple proof of Witten conjecture through localization, preprint 2005, math.AG/0508384.

    Google Scholar 

  51. T. Kimura and X. Liu, A genus-3 topological recursion relation, preprint 2005, math.DG/0502457.

    Google Scholar 

  52. M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  53. M. Kontsevich, Enumeration of rational curves via torus actions, in the Moduli Space of Curves (Texel Island, 1994), R. Dijkgraaf, C. Faber and G. van der Geer, eds., Progr. Math. vol. 129, Birkhäuser, Boston, 1995, pp. 335–368.

    Google Scholar 

  54. A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536.

    Article  MATH  MathSciNet  Google Scholar 

  55. Y.-P. Lee, Invariance of tautological equations I: Conjectures and applications, preprint 2006, math.AG/0604318v2.

    Google Scholar 

  56. Y.-P. Lee, Invariance of tautological equations II: Gromov–Witten theory, preprint 2006, available at http://www.math.utah.edu/ yplee/research/.

  57. Y.-P. Lee and R. Pandharipande, Frobenius manifolds, Gromov–Witten theory, and Virasoro constraints, book in preparation.

    Google Scholar 

  58. J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Diff. Geom. 57 (2001), 509–578.

    MATH  Google Scholar 

  59. J. Li, A degeneration formula of GW-invariants, J. Diff. Geom. 60 (2002), 199–293.

    MATH  Google Scholar 

  60. A.-M. Li and Y. Ruan, Symplectic surgery and Gromov–Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), 151–218.

    Article  MATH  MathSciNet  Google Scholar 

  61. C.-C. M. Liu, K. Liu, and J. Zhou, A proof of a conjecture of Marino-Vafa on Hodge integrals, J. Diff. Geom. 65 (2004), 289–340.

    MathSciNet  Google Scholar 

  62. E. Looijenga, On the tautological ring of \({{\cal M}}_{g}\), Invent. Math. 121 (1995), no. 2, 411–419.

    Article  MATH  MathSciNet  Google Scholar 

  63. I. Madsen and U. Tillmann, The stable mapping class group and \(Q(\mathbb{C}{\mathbb{P}}_{+}^{\infty })\), Invent. Math. 145 (2001), no. 3, 509–544.

    Article  MATH  MathSciNet  Google Scholar 

  64. I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, preprint 2002, math.AT/0212321.

    Google Scholar 

  65. M. Mirzakhani, Weil-Petersson volumes and the Witten-Kontsevich formula, preprint 2003.

    Google Scholar 

  66. S. Morita, Generators for the tautological algebra of the moduli space of curves, Topology 42 (2003), 787–819.

    Article  MATH  MathSciNet  Google Scholar 

  67. S. Morita, Cohomological structure of the mapping class group and beyond, preprint 2005, math.GT/0507308v1.

    Google Scholar 

  68. D. Mumford, Toward an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry, Vol. II, M. Artin and J. Tate ed., 271–328, Prog. Math. 36, Birk. Boston, Boston, MA, 1983.

    Google Scholar 

  69. A. Okounkov and R. Pandharipande, Gromov–Witten theory, Hurwitz numbers, and matrix models, I, math.AG/0101147.

    Google Scholar 

  70. R. Pandharipande, Three questions in Gromov–Witten theory, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 503–512, Higher Ed. Press, Beijing, 2002.

    Google Scholar 

  71. B. Riemann, Theorie der Abel’schen Funktionen, J. Reine angew. Math. 54 (1857), 115–155.

    MATH  Google Scholar 

  72. U. Tillmann, Strings and the stable cohomology of mapping class groups, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 447–456, Higher Ed. Press, Beijing, 2002.

    Google Scholar 

  73. R. Vakil, Genus 0 and 1 Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations, Trans. Amer. Math. Soc. 353 (2001), 4025–4038.

    Article  MATH  MathSciNet  Google Scholar 

  74. R. Vakil, The moduli space of curves and its tautological ring, Notices of the Amer. Math. Soc. (feature article), vol. 50, no. 6, June/July 2003, p. 647–658.

    Google Scholar 

  75. A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.

    Article  MATH  MathSciNet  Google Scholar 

  76. E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom. 1 (1991), 243–310.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA

    R. Vakil

Authors
  1. R. Vakil
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver, V6T 1Z2, BC, Canada

    Kai Behrend

  2. Department of Mathematics “Guido Castelnuovo”, University of Rome “La Sapienza”, P.le Aldo Moro 5, 00185 Rome, Italy

    Marco Manetti

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Vakil, R. (2008). The Moduli Space of Curves and Gromov–Witten Theory. In: Behrend, K., Manetti, M. (eds) Enumerative Invariants in Algebraic Geometry and String Theory. Lecture Notes in Mathematics, vol 1947. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79814-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation