Gromov–Witten theory of P1×P1×P1

We use elementary geometric techniques to exhibit an explicit equivalence between certain sectors of the Gromov–Witten theories of blowups of P1×P1×P1P1×P1×P1 and P3P3. In particular, we prove that the all genus, virtual dimension zero Gromov–Witten theory of the blowup of P3P3 at points coincides with that of the blowup at points of P1×P1×P1P1×P1×P1, for non-exceptional classes. We observe a toric symmetry of the Gromov–Witten theory of P1×P1×P1P1×P1×P1 analogous and intimately related to Cremona symmetry of P3P3. Enumerative applications are given