Tropical rational curves with first order tangency via WDVV

TL;DR

This paper develops a tropical geometry approach using WDVV equations to count rational curves with first order tangency in the plane, confirming results consistent with complex geometry.

Contribution

It introduces a tropical WDVV-based method to enumerate rational tropical curves with tangency conditions, aligning with classical complex geometry results.

Findings

Tropical counts match complex geometry results

Method computes degree d curves tangent to degree l curves

Uses tropical WDVV equations for enumeration

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.