Relative Gromov-Witten and maximal contact conics

We discuss some properties of the relative Gromov–Witten invariants counting rational curves with maximal contact order at one point. We compute the number of Cayley’s sextactic conics to any smooth plane curve $Y$. In particular, we compute the contribution, from double covers of inflectional lines, to a certain degree $2$ relative Gromov–Witten invariant relative to $Y$.