Enumerative Geometry of Del Pezzo Surfaces

TL;DR

This paper establishes a connection between tropical geometry and Lagrangian Floer theory for del Pezzo surfaces, providing explicit calculations for the projective plane that support existing conjectures.

Contribution

It demonstrates an equivalence between superpotentials from tropical geometry and Floer theory, with explicit examples for the projective plane.

Findings

Confirmed folklore conjecture for the projective plane

Established equivalence between superpotential definitions

Provided explicit calculations for del Pezzo surfaces

Abstract

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.