Node polynomials for curves on surfaces

Abstract

This text is a presentation of a set of formulae, first found by Vainsencher (for $\hat{I}´\le 6$) and shortly after improved by Kleiman and Piene, counting $\hat{I}´$-nodal curves in a complete linear system on a smooth surface, if $\hat{I}´\le 8$ and the corresponding line bundle is sufficiently positive. We also discuss a complement by Qviller, and related results due to Kazarian, Ohmoto, and others.