Rational cuspidal curves on del-Pezzo surfaces

Abstract

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler class computation on the moduli space of curves. A topological method is employed in computing the contribution of the degenerate locus to this Euler class.