Gromov–Witten Theory of Blowups of Toric Threefolds

We use toric symmetry and blowups to study relationships in the Gromov– Witten theories of P3 and P1×P1×P1. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of P3 and P1×P1×P1 by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of P3 and P1×P1×P1. Enumerative consequences are discussed.