In this paper, we discuss the scheme of enumerating the singular holomorphic curves in a linear system on an algebraic surface. Or approach is based on the usage of the family Seiberg-Witten invariant and tools from differential topology and algebraic geometry.

In particular, one shows that the number of δ-nodes nodal curves in a generic δ dimensional sub-linear system can be expressed as a universal degree δ polynomial in terms of the four basic numerical invariants of the linear system and the algebraic surface. The result enables us to study in detail the structure of these enumerative invariants.