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Tropical rational curves with first order tangency via WDVV

Anantadulal Paul, Aditya Subramaniam

TL;DR

This work develops a tropical WDVV framework to count rational tropical plane curves of degree $d$ tangent to a fixed tropical curve of degree $l$ and passing through $3d-2$ generic points. By tropicalizing the classical WDVV degeneration and introducing the map $\tilde{\pi}$, the authors derive a Kontsevich-type recursion for $N_d^{\mathsf{T}_1}(l)$, incorporating intermediate counts $N_d^{\mathsf{T}_0}$ and $n_d$. The tropical counts are shown to coincide with complex tangency numbers via Mikhalkin’s correspondence, validating the tropical method as a robust tool for tangency problems. The results provide a concrete computational framework and connect tropical geometry with classical enumerative geometry, enabling broader exploration of tangency phenomena in the tropical setting.

Abstract

In this article, we study the tropical counterpart of the enumeration of rational curves in $\mathbb{CP}^2$ with first order tangency. We use the tropical analogue of the WDVV technique to compute rational tropical plane curves of degree $d$ tangent to a degree $l$ tropical plane curve and passing through $3d-2$ points in general position. As Mikhalkin's correspondence suggests, our numbers agree with earlier results on tangency in complex geometry.

Tropical rational curves with first order tangency via WDVV

TL;DR

This work develops a tropical WDVV framework to count rational tropical plane curves of degree tangent to a fixed tropical curve of degree and passing through generic points. By tropicalizing the classical WDVV degeneration and introducing the map , the authors derive a Kontsevich-type recursion for , incorporating intermediate counts and . The tropical counts are shown to coincide with complex tangency numbers via Mikhalkin’s correspondence, validating the tropical method as a robust tool for tangency problems. The results provide a concrete computational framework and connect tropical geometry with classical enumerative geometry, enabling broader exploration of tangency phenomena in the tropical setting.

Abstract

In this article, we study the tropical counterpart of the enumeration of rational curves in with first order tangency. We use the tropical analogue of the WDVV technique to compute rational tropical plane curves of degree tangent to a degree tropical plane curve and passing through points in general position. As Mikhalkin's correspondence suggests, our numbers agree with earlier results on tangency in complex geometry.

Paper Structure

This paper contains 6 sections, 8 theorems, 30 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview of our result
  3. Plane tropical curves
  4. Intermediate intersection numbers
  5. Kontsevich type formula for tangency
  6. Verification with existing results

Key Result

Proposition 3.3

For any positive integer $n$ and combinatorial type $\alpha$, the space $\mathcal{M}^{\alpha}_n$ (respectively $\mathcal{M}^{\alpha}_{\bigtriangleup, n}$) is naturally an (unbounded) open convex polyhedron in a real vector space of expected dimension

Figures (5)

  • Figure 1: Two distinct transverse intersections limiting to a first order tangency
  • Figure 2: Tropical curves of types (A), (B), (C), and (D)
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (23)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Example 3.5
  • Lemma 4.1
  • proof
  • Definition 5.1
  • Definition 5.2
  • ...and 13 more