Abstract
Abstract In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, 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 ℙ3 with one cusp (for δ ≤ 2).
Highlights
A classical question in enumerative algebraic geometry is: Question
In this paper we obtain a formula for the number of rational degree d curves in P having a cusp, whose image lies in a P and that passes through r lines and s points
This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in P, which has been studied earlier by Z
Summary
A classical question in enumerative algebraic geometry is: Question. What is Nd, the number of rational (genus zero) degree d curves in P that pass through d − generic points?. Zinger ([16], [17] and [15]) These results have been generalized to other surfaces (such as P × P ) by J. Ran ([13]), and more recently a solution to this question in any genus has been obtained by Y. A natural generalization of problems in enumerative geometry (where one studies curves inside some xed ambient surface such as P ) is to consider a family version of the same problem. Laarakker ([10]) where they study the enumerative geometry of nodal curves in a moving family of surfaces
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