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

TL;DR

This paper surveys how tropical geometry techniques are used to count plane curves over complex, real, and quadratically enriched fields, unifying various enumerative invariants.

Contribution

It provides an overview of tropical methods applied to complex, real, and quadratically enriched plane curve counting problems, highlighting recent advances.

Findings

Tropical geometry enables parallel computation of complex and real invariants.

Quadratically enriched enumerative geometry unifies counts over multiple fields.

Tropical methods facilitate the computation of quadratically enriched invariants.

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.