A tree formula for the ellipsoidal superpotential of the complex projective plane

TL;DR

This paper derives a closed-form combinatorial formula for counting specific rational plane curves with cusps in the complex projective plane, advancing understanding of the ellipsoidal superpotential's properties.

Contribution

It introduces a novel tree-based formula for the ellipsoidal superpotential, linking geometric curve counts to combinatorial structures.

Findings

Provides a sum-over-trees formula for curve counts

Clarifies the nonvanishing properties of the superpotential

Enhances understanding of the combinatorial nature of the superpotential

Abstract

The ellipsoidal superpotential of the complex projective plane can be interpreted as a count of rigid rational plane curves of a given degree with one prescribed cusp singularity. In this note we present a closed formula for these counts as a sum over trees with certain explicit weights. This is a step towards understanding the combinatorial underpinnings of the ellipsoidal superpotential and its mysterious nonvanishing and nondecreasing properties.