Title: Counting singular curves on algebraic surfaces
Abstract: The Göttsche conjecture, now a theorem, states that the number of nodal curves on a surface can be expressed universally as a polynomial in four topological invariants. Kleiman and Piene have formulated a family version of this conjecture. It states that for a relative effective divisor on a family of surfaces, the number of nodal fibres of the divisor can be expressed universally in the intersection numbers of the surface and the divisor. The conjecture can be proven, using the BPS-calculus of Pandharipande and Thomas, and the language of constructible functions.