Refined broccoli invariants

TL;DR

This paper introduces a new tropical enumerative invariant called the refined broccoli invariant, which generalizes existing invariants and connects to tropical Gromov-Witten invariants, providing new insights and potential generalizations.

Contribution

It defines the refined broccoli invariant, proves its independence from point configurations, and links it to tropical Gromov-Witten invariants, advancing the understanding of tropical enumerative geometry.

Findings

Refined broccoli invariant is independent of point configuration.

Specializes to tropical Gromov-Witten invariants at y=1.

Provides a new perspective on broccoli invariants.

Abstract

We introduce a tropical enumerative invariant depending on a variable y which generalizes the tropical refined Severi degree. We show that this refined broccoli invariant is indeed independent of the point configuration, and that it specializes to a tropical descendant Gromov-Witten invariant for y=1 and to the corresponding broccoli invariant for y=-1. Furthermore, we define tropical refined descendant Gromov-Witten invariants which equal the corresponding refined broccoli invariants giving a new insight to the nature of broccoli invariants. We discuss various possible generalizations, e.g. to refinements of bridge curves and Welschinger curves.