Rational cuspidal curves in a moving family of ℙ 2

In this paper we obtain a formula for the number of rational degree d curves in ℙ 3 having a cusp, whose image lies in a ℙ 2 and that passes through r lines and s points (where r + 2 s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ 2 , which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ 3 with one cusp (for δ ≤ 2).