Counting curves with local tangency constraints

We construct invariants of any semipositive symplectic manifold which count rational curves satisfying multibranched tangency constraints to a local divisor. We also construct analogous invariants counting punctured curves with negative ends on a small skinny ellipsoid, and we prove that these counts coincide at least in dimension four. We then give a formula describing how tangency constraints arise as point constraints are pushed together, and we use this to recursively compute all invariants in dimension four in terms of Gromov-Witten invariants of blowups.