Ritwik Mukherjee, Rahul Kumar Singh
February 24, 2021
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 (), R. Pandharipande () and A. Zinger (). 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 () and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali (). 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 (), where they compute the characteristic number of δ-nodal planar curves in ℙ 3 with one cusp (for δ ≤ 2).