manuscripta mathematica

, Volume 129, Issue 3, pp 293–335

Tropical descendant Gromov–Witten invariants

Open Access


We define tropical Psi-classes on\({\mathcal{M}_{0,n}(\mathbb{R}^2, d)}\) and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin’s lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions.

Mathematics Subject Classification (2000)

Primary 14N35 52B20 


  1. 1.
    Allermann, L., Rau, J.: First steps in tropical intersection theory. Preprint. math. AG/0709.3705Google Scholar
  2. 2.
    Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic Geometry, Proceedings of the Summer Research Institute Santa Cruz 1995. Proc. Symp. Pure Math., vol. 62, part 2, pp. 45–96 (1997)Google Scholar
  3. 3.
    Gross, M.: Mirror Symmetry for \({\mathbb{P}^2}\) and Tropical Geometry. Preprint. math.AG/0903. 1378Google Scholar
  4. 4.
    Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compositio Mathematica (to appear). math.AG/0708.2268Google Scholar
  5. 5.
    Gathmann A., Markwig H.: Kontsevich’s formula and the WDVV equations in tropical geometry. Adv. Math. 217, 537–560 (2008) math.AG/0509628MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kock, J.: Notes on Psi classes.
  7. 7.
    Kerber, M., Markwig, H.: Intersecting Psi-classes on tropical M 0,n. Preprint. math.AG/0709.3953Google Scholar
  8. 8.
    Mann, B.: An equivalent condition for the reducibility of tropical curves. Preprint, University of Michigan, Ann Arbor (in preparation)Google Scholar
  9. 9.
    Markwig, H.: The enumeration of plane tropical curves. PhD thesis, TU Kaiserslautern (2006)Google Scholar
  10. 10.
    Mikhalkin G.: Enumerative tropical geometry in \({\mathbb{R}^2}\). J. Am. Math. Soc. 18, 313–377 (2005) math.AG/0312530MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mikhalkin, G.: Tropical geometry and its applications. In: Sanz-Sole, M., et al. (eds.) Invited Lectures, vol. II. Proceedings of the ICM Madrid, pp. 827–852 (2006). math.AG/0601041Google Scholar
  12. 12.
    Mikhalkin, G.: Moduli spaces of rational tropical curves. Preprint. math.AG/0704.0839 (2007)Google Scholar
  13. 13.
    Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. Idempotent Mathematics and Mathematical Physics, Proceedings Vienna (2003). math/0306366Google Scholar
  14. 14.
    Speyer, D., Sturmfels, B.: Tropical mathematics. Preprint. math.CO/0408099 (2004)Google Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Courant Research Center “Higher Order Structures in Mathematics”Georg-August-Universität GöttingenGöttingenGermany
  2. 2.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany

Personalised recommendations