The number of plane conics 5-fold tangent to a given curve

Given a general plane curve Y of degree d, we compute the number n_d of irreducible plane conics that are 5-fold tangent to Y. This problem has been studied before by Vainsencher using classical methods, but it could not be solved there because the calculations received too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number n_d in terms of relative Gromov-Witten invariants that can then be directly computed. As an application, we consider the K3 surface given as the double cover of P^2 branched along a sextic curve. We compute the number of rational curves in this K3 surface in the homology class that is the pull-back of conics in P^2, and compare this number to the corresponding Yau-Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class.