Gromov-Witten Theory of P^1xP^1xP^1

TL;DR

This paper establishes an explicit equivalence between certain Gromov-Witten theories of blowups of P^1xP^1xP^1 and P^3, revealing symmetries and providing enumerative applications.

Contribution

It demonstrates a geometric equivalence between sectors of Gromov-Witten theories of specific blowups of P^1xP^1xP^1 and P^3, highlighting symmetries and applications.

Findings

Gromov-Witten theories of blowups of P^3 and P^1xP^1xP^1 coincide in certain sectors

Identifies a toric symmetry related to Cremona symmetry

Provides enumerative applications of the theory

Abstract

We use elementary geometric techniques to exhibit an explicit equivalence between certain sectors of the Gromov-Witten theories of blowups of P^1xP^1xP^1 and P^3. In particular, we prove that the all genus, virtual dimension zero Gromov-Witten theory of the blowup of P^3 at points coincides with that of the blowup at points of P^1xP^1xP^1, for non-exceptional classes. We observe a toric symmetry of the Gromov-Witten theory of P^1xP^1xP^1 analogous and intimately related to Cremona symmetry of P^3. Enumerative applications are given.