The local Gromov-Witten invariants of configurations of rational curves

We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. We first transform this from a problem involving local Gromov-Witten invariants to one involving global or ordinary invariants. We do so by expressing the local invariants of a configuration of curves in terms of ordinary Gromov-Witten invariants of a blowup of $\mathbb{CP}$$^{3}$ at points. The Gromov-Witten invariants of a blowup of $\mathbb{CP}$$^{3}$ along points have a symmetry, which arises from the geometry of the Cremona transformation, and transforms some difficult to compute invariants into others that are less difficult or already known. This symmetry is then used to compute the global invariants.