Tropical methods in enumerative geometry

This semester we will examine an approach to enumerative geometry via tropical geometry, especially related to the case of enumerative problems over the reals. We will present some of the history and recent developments in this area. The ultimate goal is to learn how to apply these methods to more general fields using methods of motivic homotopy theory, but due to time constraints, this aspect will not appear in the seminar program except perhaps during our discussions.

The seminar will take place online via Zoom; we meet on Tuesdays, 16-18 Uhr. If you are interested in attending the seminar or giving one of the lectures, please contact me (Marc Levine) at

Previous Motives Seminars

Follow the Teaching link at the top of this page to find links to previous motives seminars.

Program: Refined Tropical Invariants

Tropical enumerative invariants are invariants associated to toric surfaces that are defined in a purely combinatorial manner and coincide with classical complex and real geometric ones. We will present the definition and properties of the classical and tropical invariants, and the equalities they satisfy.

Lecture 1 (13.04-Andrés Jaramillo Puentes) Introduction to the seminar
Lecture 1 Slides
Lecture 1 Video

Lecture 2 (20.04-Michele Ancona) Kontsevich Formula for enumeration of Rational Plane Curves [KM]
Lecture 2 Notes
Lecture 2 Video

The numbers of complex plane rational curves of degree d through \(3d-1\) general given points can be computed recursively with Kontsevich's formula. The proof is obtained by degenerating the configuration of points to a non-generic configuration so that the curves split into curves of smaller degree and uses a map to the moduli space \(\bar{M}_{0,4}\), in which the cohomology classes of any two points is the same, in particular of two points in the boundary \( \bar{M}_{0,4}\setminus M_{0,4}\). The goal of the talk is to survey this construction in detail and understand the combinatorial coefficients of the formula.

Lecture 3 (27.04-Sabrina Pauli) Caporasso-Harris recursion formula for plane curves of positive genus [CH]
Lecture 3 Notes
Lecture 3 Video

Caporaso and Harris have found a nice way to compute the numbers \(N(d,g)\) of complex plane curves of degree \(d\) and genus \(g\) through \(3d+g-1\) general points with the help of relative Gromov-Witten invariants. They use the same idea of degenerating the configurations of points towards a non-generic configuration (e.g. putting three points over a fixed line, in which case the GW invariants would be relative to such) and forcing the counted curves to split into products of curves of lesser degree and genus. The goal of the talk is to understand the use of these relative invariants in exploiting the argument for rational curves.

Lecture 4 (04.05-Marc Levine) Welschinger invariants for enumeration of real curves [W]
Lecture 4 Notes
Lecture 4 Video

Over the reals, the number of curves of fixed degree passing through a generic configuration of points is not invariant (i.e. different configurations yields different amounts of curves). Welschinger introduced coefficients for the rational curves so that counting the curves with this coefficients is an invariant. The goal of the talk is to understand the proof of invariance (at least for the real projective plane); the new families of invariants that appears depend on the number of complex conjugate pairs of the configuration of points. You should also mention the formulas satisfied by this invariant for the blow up of surfaces.

Lecture 5 (11.05-Alessandro D'Angelo) Introduction to Tropical Geometry I. [BIMS] [IMS]
Lecture 5 Slides
Lecture 5 Video

The goal of this talk is to introduce the tropicalization map for a plane curve. The first part is an introduction of tropical curves in \(\mathbb{R}^2\) (tropical algebra, zeros of a polynomial, hypersurfaces, balancing condition, degree, genus). A second part should give an introduction to amoebas and the tropical map.

Lecture 6 (18.05-Maria Yakerson) Introduction to Tropical Geometry II. [BIMS] [IMS] [M]
Lecture 6 Notes
Lecture 6 Video

Having introduced tropical curves in Lecture 5, we would like to explore some of their properties. The goal of the talk is to introduce the Newton polytope of a polynomial, its subdivision and its duality with respect to the tropical curve; intersection multiplicity and Bezout's theorem; and the floor diagram. [BIMS] [IMS] [M]

Lecture 7 (25.05-Heng Xie) Tropical Gromov-Witten invariants [M]
Lecture 7 Slides
Lecture 7 Video

Given a generic configuration of points in \(\mathbb{R}^2\), counting the tropical curves of fixed degree and genus with a complex (or real) multiplicity yields an invariant. The goal of this talk is introduce the complex and real multiplicities associated to a tropical curve and to survey the invariance with respect to the configuration of points.

Lecture 8 (01.06-Manh Toan Nguyen) Mikhalkin correspondence Theorem [M]
Lecture 8 Notes
Lecture 8 Video

The invariant obtained by counting the tropical curves of fixed degree and genus with a complex (or real) curve passing through a generic configuration of points coincide with the Gromov-Witten invariant (or Welschinger invariant, respectively). The goal of this talk is to survey the proof of this equality.

Lecture 9 (08.06-Longting Wu) Block-Göttsche refined invariants [IM]
Lecture 9 Notes
Lecture 9 Video

Block-Göttsche introduced a quantum multiplicity for tropical curves. It's a symmetric Laurent polynomial that specializes to the Mikhalkin complex and real multiplicites. The goal of this talk is to introduce this Block-Göttsche multiplicity, survey the proof of the new invariant defined by counting the tropical curves of given genus and degree passing through a generic configuration of points counted with this quantum multiplicity, and to show examples, presenting its specialization to the Gromov-Witten and Welschinger invariants.

Lecture 10 (15.06-Viktor Kleen) Broccoli invariants [GMS]
Lecture 10 Notes
Lecture 10 Video

Broccoli curves are purely combinatorial objects that were introduced in order to prove the invariance of the corresponding Welschinger numbers and also to find formulas to compute them. The goal of this talk is to introduce them, survey the proof of their invariance and explain the relation with Welschinger numbers.

Lecture 11 (22.06-Tariq Syed) Refined broccoli invariants [GS16]
Lecture 11 Slides
Lecture 11 Video

The goal of this talk is to introduce the refined broccoli invariants obtained by counting the broccoli curves with the quantum multiplicity, and to show examples of this invariant specializing to Welschinger invariants corresponding to configuration of points with complex conjugate pairs.

Lecture 12 (29.06-Sabrina Pauli) Caporasso-Harris formula for refined invariants [BG][GM]
Lecture 12 Notes
Lecture 12 Video

The Caporasso-Harris formula defines a recursion that allow us to calculate positive genus invariants. The goal of this talk is to understand a tropical version (with quantum multiplicities) of this recursion using relative invariants and floor diagrams, and to present some computations in low degree.

Lecture 13 (06.07-Andrés Jaramillo Puentes) Node polynomials and the Göttsche conjecture [AB][GS12][BJP]
Lecture 13 Slides
Lecture 13 Video

Block and Göttsche showed that coefficients of fixed degree of the refined node polynomials (or Severi degrees) are polynomial with respect to the degree of the counted curves. Ardila and Block showed that this phenomena occurs for node polynomials when counting curves on other toric surfaces (like the Hirzebruch surfaces). Brugallé and Jaramillo Puentes showed that while the Gromov-Witten invariants for a fixed genus are exponential, coefficient of fixed codegree are polynomial with respect to the degree or with respect the Newton polygon of the toric surface. The goal of this talk is to explain how these polynomial properties are generalizations of the Göttsche conjecture for node polynomials.


[AB] F. Ardila, F. Block. Universal Polynomials for Severi Degrees of Toric Surfaces arXiv:1012.5305

[BG] F. Block and L. Göttsche. Refined curve counting with tropical geometry. Compositio Mathematica, 152(1):115–151, 2016.

[BJP] E. Brugallé, A. Jaramillo Puentes. Polynomiality properties of tropical refined invariants.

[BIMS] E. Brugallé, I. Itenberg, G. Mikhalkin, and K. Shaw. Brief introduction to tropical geometry, arXiv:1502.05950 Proceedings of Gokova Geometry/Topology conference 2014, International Press (2015), 1 - 75.

[BM] E. Brugallé,  G. Mikhalkin. Floor decompositions of tropical curves : the planar case arXiv:0812.3354

[CH] L. Caporaso and J. Harris, Counting plane curves of any genus, Invent. math. 131  (1998), 345–392

[GM] A. Gathmann, H. Markwig. The Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry arXiv:math/0504392

[GMS] A. Gathmann, H. Markwig, and F. Schroeter, Broccoli curves and the tropical invariance of Welschinger numbers Adv. Math. 240 (2013), 520–574.

[GS12] L. Göttsche, V. Shende Refined curve counting on complex surfaces arXiv:1208.1973

[GS16] L. Göttsche, F. Schroeter. Refined broccoli invariants arXiv:1606.09631

[IM] I. Itenberg, G. Mikhalkin. On Block-Goettsche multiplicities for planar tropical curves arXiv:1201.0451 , Intern. Math. Res. Notices 23 (2013), 5289 – 5320

[IMS] I. Itenberg, G. Mikhalkin and E. Shustin. Tropical Algebraic Geometry, Birkhäuser, Oberwolfach Seminars Series, Vol. 35, 2007.

[KM] M. Kontsevich and Y. I. Manin. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164 (1994),  525–562. (hep-th/9402147).

[M] G. Mikhalkin. Enumerative tropical algebraic geometry in \(\mathbb{R}^2\), arXiv:math/0312530  Journal of the AMS.

[W] J.Y. Welschinger. Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. Invent. Math., 162(1):195–234, 2005