Enumerating singular curves on surfaces

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basic Chern numbers. To justify the enumeration, we make a rudimentary classification of the types of singularities using Enriques diagrams, obtaining results like Arnold’s. We show that the curves in question do, in fact, appear with multiplicity 1 using the versal deformation space, Shustin’s codimension formula, and Gotzmann’s regularity theorem. Finally, we relate our work to Vainsencher’s work with up to seven nodes.