abstract: I'll describe joint work with B. Hassett in which we use the degeneration formula in relative Gromov-Witten theory to explain and generalize a formula first discovered by Takahashi and proven by Gathmann: the virtual number of degree d rational curves in the total space of the canonical bundle of P2 is equal to a simple multiple of the virtual number of degree d immersions of A1 into the complement of a smooth plane cubic.