In [R2] we showed how elementary considerations involving geometry on ruled surfaces may be used to obtain recursive enumerative formulae for rational plane curves. Here we show how similar considerations may be used to obtain further enumerative formulae, as follow. First some notation. As usual we denote by Ngd the number of irreducible plane curves of degree d and genus g through 3d+g−1 general points. Also, we denote by Ngd→ (resp. Ngd×) the number of such curves passing through general points A1, …, A3d+g−2 and having a given tangent direction (resp. a node) at A1. As is well known and easy to see, we have

formula here

For any d, g, these numbers are computed in [R1] as part of a more general recursive procedure. For N0d, N1d, relatively simple recursions have been given by Kontsevich–Manin (see [FP]) and Eguchi–Hori–Xiong–Getzler (see [P1]), respectively.