  Let S be a complex smooth projective surface and L be a line bundle on S. For
any given collection of isolated topological or analytic singularity types, we
show the number of curves in the linear system |L| with prescribed
singularities is a universal polynomial of Chern numbers of L and S, assuming L
is sufficiently ample. Moreover, we define a generating series whose
coefficients are these universal polynomials and discuss its properties. This
work is a generalization of Gottsche's conjecture to curves with higher
singularities.
