Gromov-Witten theory of P1 x P1 x P1

We use elementary geometric techniques to exhibit an explicit equivalence between certain sectors of the Gromov-Witten theories of blowups of P-1 x P-1 x P-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-1 x P-1 x P-1, for non-exceptional classes. We observe a toric symmetry of the Gromov-Witten theory of P-1 xP(1) xP(1) analogous and intimately related to Cremona symmetry of P-3. Enumerative applications are given. (C) 2016 Elsevier B.V. All rights reserved.