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Tropical Methods for Counting Plane Curves -- Complex, Real and Quadratically Enriched

Andrés Jaramillo Puentes, Hannah Markwig, Sabrina Pauli, Felix Röhrle

TL;DR

This survey unifies tropical methods for counting plane rational curves across complex, real, and quadratically enriched settings, foregrounding Mikhalkin’s correspondence that equates complex counts with tropical counts. It explicates both the standard complex and real counts, and then extends to quadratically enriched counts valued in the Grothendieck–Witt ring, showing a tropical correspondence that subsumes these variants. Central technical tools include dual Newton subdivisions, tropicalization via Puiseux series and Kapranov’s theorem, and lattice-path algorithms whose quadratically enriched multiplicities reproduce the enriched invariants. The work also outlines algorithmic approaches, notably lattice-path methods, and discusses extensions to point conditions defined over quadratic field extensions, highlighting the potential to derive multiple real and complex counts from a single tropical computation. Overall, the paper emphasizes tropical geometry as a powerful framework for computing and understanding refined enumerative invariants across base fields, with practical implications for both theory and computation.

Abstract

Since the first famous correspondence theorem by Mikhalkin appeared in 2005, tropical geometry has allowed a parallel treatment of real and complex counting problems. A prime example are the genus 0 Gromov-Witten invariants of the plane which count rational plane curves of degree d satisfying point conditions and their real counterpart, the Welschinger invariants, which both can be determined using tropical methods. Remarkably, the tropical computation of the two types of invariants works entirely in parallel. Recently, quadratically enriched enumerative geometry enables us to combine such real and complex counts under one roof, providing a simultaneous approach which can also be used for counts over other fields. Tropical geometry is a successful tool for the study and computation of such quadratically enriched enumerative invariants, too. In this survey, we provide an overview of tropical methods for plane curve counting problems over the real and complex numbers, and the new quadratically enriched counts.

Tropical Methods for Counting Plane Curves -- Complex, Real and Quadratically Enriched

TL;DR

This survey unifies tropical methods for counting plane rational curves across complex, real, and quadratically enriched settings, foregrounding Mikhalkin’s correspondence that equates complex counts with tropical counts. It explicates both the standard complex and real counts, and then extends to quadratically enriched counts valued in the Grothendieck–Witt ring, showing a tropical correspondence that subsumes these variants. Central technical tools include dual Newton subdivisions, tropicalization via Puiseux series and Kapranov’s theorem, and lattice-path algorithms whose quadratically enriched multiplicities reproduce the enriched invariants. The work also outlines algorithmic approaches, notably lattice-path methods, and discusses extensions to point conditions defined over quadratic field extensions, highlighting the potential to derive multiple real and complex counts from a single tropical computation. Overall, the paper emphasizes tropical geometry as a powerful framework for computing and understanding refined enumerative invariants across base fields, with practical implications for both theory and computation.

Abstract

Since the first famous correspondence theorem by Mikhalkin appeared in 2005, tropical geometry has allowed a parallel treatment of real and complex counting problems. A prime example are the genus 0 Gromov-Witten invariants of the plane which count rational plane curves of degree d satisfying point conditions and their real counterpart, the Welschinger invariants, which both can be determined using tropical methods. Remarkably, the tropical computation of the two types of invariants works entirely in parallel. Recently, quadratically enriched enumerative geometry enables us to combine such real and complex counts under one roof, providing a simultaneous approach which can also be used for counts over other fields. Tropical geometry is a successful tool for the study and computation of such quadratically enriched enumerative invariants, too. In this survey, we provide an overview of tropical methods for plane curve counting problems over the real and complex numbers, and the new quadratically enriched counts.
Paper Structure (21 sections, 11 theorems, 23 equations, 11 figures, 1 table)

This paper contains 21 sections, 11 theorems, 23 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. What is a tropical plane curve?
  4. Tropical plane curves as piecewise linear graphs dual to Newton subdivisions
  5. Tropicalization of plane curves
  6. Complex enumerative geometry and correspondences
  7. Counting complex plane curves
  8. Counting tropical plane curves
  9. The correspondence theorem
  10. Welschinger invariants
  11. Counting real plane curves
  12. The tropical count of Welschinger invariants
  13. Quadratically enriched counts through points defined over the ground field
  14. Background on quadratically enriched counts
  15. Quadratically enriched counts of rational plane curves satisfying point conditions defined over the ground field $k$
  16. ...and 6 more sections

Key Result

Theorem 2.14

Let $C\subset (k\{\!\{t\}\!\}^\times)^2$ be a curve, where $k$ is algebraically closed of characteristic zero. Then the closure in the Euclidean topology of the tropicalization $\overline {\operatorname{Trop}(C)}$ is a piecewise integer affine linearly embedded graph in $\mathbb{R}^2$ which satisfie

Figures (11)

  • Figure 1: Two tropical plane curves.
  • Figure 2: The dual Newton subdivision of the two tropical plane curves depicted in Figure \ref{['fig-planecurves']}.
  • Figure 3: A tropical plane curve parametrized by an abstract graph, i.e. a tropical stable map.
  • Figure 4: A local picture of a tropical stable map: the marked end $x_1$ meets the point condition $p_1$.
  • Figure 5: The tropicalization of a line is the image of a tropical stable map of degree $1$.
  • ...and 6 more figures

Theorems & Definitions (55)

  • Example 2.1
  • Definition 2.2: Rational abstract tropical curve
  • Definition 2.3: Rational tropical stable map to $\mathbb{R}^2$
  • Example 2.4
  • Remark 2.5: Drawing convention
  • Remark 2.6: Dual Newton subdivision
  • Remark 2.7: Point conditions
  • Definition 2.8: Degree
  • Example 2.9
  • Remark 2.10: Other degrees
  • ...and 45 more