Rocky Mountain Journal of Mathematics

Intersections on tropical moduli spaces

Johannes Rau

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a ``boundary'' divisor, and we prove general tropical versions of the WDVV, respectively, topological recursion equations (under some assumptions). As a direct application, we prove that, for the toric varieties $\PP ^1$, $\PP ^2$, $\PP ^1 \times \PP ^1$ and with $\Psi $-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\PP ^2$ in \cite {MR08}). Our approach uses tropical intersection theory and unifies and simplifies some parts of the existing tropical enumerative geometry (for rational curves).

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 2 (2016), 581-662.

Dates
First available in Project Euclid: 26 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1469537479

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-581

Mathematical Reviews number (MathSciNet)
MR3529085

Zentralblatt MATH identifier
1379.14035

Subjects
Primary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Keywords
Tropical Gromov-Witten theory tropical intersection theory tropical geometry

Citation

Rau, Johannes. Intersections on tropical moduli spaces. Rocky Mountain J. Math. 46 (2016), no. 2, 581--662. doi:10.1216/RMJ-2016-46-2-581. https://projecteuclid.org/euclid.rmjm/1469537479


Export citation

References

  • Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), 633–670.
  • Lars Allermann and Johannes Rau, Tropical rational equivalence on $\RR^r$, preprint, arxiv:0811.2860.
  • Renzo Cavalieri, Paul Johnson and Hannah Markwig, Tropical Hurwitz numbers, J. Alg. Comb. 32 (2010), 241–265, also at arxiv:0804.0579.
  • Marina Franz, The tropical Kontsevich formula for toric surfaces, Ph.D. thesis, TU Kaiserslautern, 2008.
  • Marina Franz and Hannah Markwig, Tropical enumerative invariants of $\FF_0$ and $\FF_2$, Adv. Geom. 11 (2011), 49–72.
  • William Fulton and Rahul Pandharipande, Notes on stable maps and quantum cohomology, Proc. Symp. Pure Math. 62 (1997), 45–96.
  • William Fulton and Bernd Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), 335–353.
  • Andreas Gathmann, Michael Kerber and Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Comp. Math. 145 (2009), 173–195.
  • Andreas Gathmann and Hannah Markwig, Kontsevich's formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), 537–560.
  • Andreas Gathmann and Eva-Maria Zimmermann, The WDVV equations in tropical geometry, in preparation.
  • Matthias Herold, Intersection theory of the tropical moduli spaces of curves, Ph.D. thesis, TU Kaiserslautern, 2007.
  • Eric Katz, A tropical toolkit, Expo. Math. 27 (2009), 1–36.
  • ––––, Tropical intersection theory from toric varieties, preprint, arxiv:0907.2488.
  • Michael Kerber and Hannah Markwig, Counting tropical elliptic plane curves with fixed $j$-invariant, Comm. Math. Helv. 84 (2009), 387–427.
  • ––––, Intersecting Psi-classes on tropical $M_{0,n}$, Int. Math. Res. Not. 2009, 221–240.
  • Joachim Kock and Israel Vainsencher, An Invitation to quantum cohomology, Progr. Math. 249, 2007.
  • Hannah Markwig and Johannes Rau, Tropical descendant Gromov-Witten invariants, Manuscr. Math. 129 (2009), 293–335.
  • Grigory Mikhalkin, Enumerative tropical geometry in $\RR^2$, J. Amer. Math. Soc. 18 (2005), 313–377.
  • ––––, Tropical geometry and its applications, Int. Cong. Math. 2 (2006), 827–852.
  • ––––, Moduli spaces of rational tropical curves, preprint, arxiv:0704.0839.
  • David Speyer and Bernd Sturmfels, Tropical Grassmannians, Adv. Geom. 4 (2004), 389–411.
  • Eva-Maria Zimmermann, Generalizations of the tropical Kontsevich formula to higher dimensions, Ph.D. thesis, TU Kaiserslautern, 2007.