Node polynomials for families: results and examples

Abstract

We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm G\"ottsche's conjecture about plane curves of low degree. Second, we justify Vainsencher's enumeration of irreducible six-nodal plane curves on a general quintic threefold in four-space. Third, we supplement Bryan and Leung's enumeration of nodal curves in a given homology class on an Abelian surface of Picard number one.