Cremona symmetry in Gromov-Witten theory

TL;DR

This paper uncovers a Cremona symmetry in Gromov-Witten theory related to toric geometry, simplifying computations of invariants and providing a new proof of the uniqueness of the rational normal curve.

Contribution

It introduces a novel Cremona symmetry in Gromov-Witten theory using permutohedron toric geometry, enabling easier computation of invariants.

Findings

Established a Cremona symmetry in Gromov-Witten invariants.

Provided a new proof of the uniqueness of the rational normal curve.

Linked symmetry to toric geometry and degeneration techniques.

Abstract

We establish the existence of a symmetry within the Gromov-Witten theory of $\mathbb{CP}^n$ and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants.