Refined curve counting with tropical geometry

TL;DR

This paper introduces a tropical geometry approach to refined Severi degrees, providing polynomial formulas in degree and refinement variable, and confirming conjectural equivalences for small delta values.

Contribution

It offers the first tropical geometric description of refined Severi degrees and proves their polynomial nature, confirming conjectures for delta up to 10.

Findings

Refined Severi degrees are polynomials in degree d and variable y for large d.

The tropical and floor diagram counts agree with G"ottsche and Shende's refinements for delta ≤ 10.

The approach confirms conjectural equivalences between different refined invariants.

Abstract

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with delta nodes. Recently, G\"ottsche and Shende gave two refinements of Severi degrees, polynomials in a variable y, which are conjecturally equal, for large d. At y = 1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed delta, the refined Severi degrees are polynomials in d and y, for large d. As a consequence, we show that, for delta <= 10 and all d, both refinements of G\"ottsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.