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In this paper we obtain an explicit formula for the number of curves in acompact complex surface $X$ (passing through the right number of genericpoints), that has up to one node and one singularity of codimension $k$,provided the total codimension is at most $7$. We use a classical fact fromdifferential topology: the number of zeros of a generic smooth section of avector bundle $V$ over $M$, counted with signs, is the Euler class of $V$evaluated on the fundamental class of $M$.

Counting curves on a general linear system with up to two singular points