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In this paper, we develop a systematic approach to enumerate curves with acertain number of nodes and one further singularity which maybe moredegenerate. As a result, we obtain an explicit formula for the number of curvesin a sufficiently ample linear system, passing through the right number ofgeneric points, that have $\delta$ nodes and one singularity of codimension$k$, for all $\delta+k \leq 8$. In particular, we recover the formulas forcurves with upto six nodal points obtained by Vainsencher. Moreover, all thecodimension seven numbers we have obtained agree with the formulas obtained byKazarian. Finally, in codimension eight, we recover the formula of A.Weber,M.Mikosz and P.Pragacz for curves with one singular point and we also recoverthe formula of Kleiman and Piene for eight nodal curves. All the othercodimension eight numbers we have obtained are new.

Counting curves in a linear system with upto eight singular points