(Hilb m V), which we use to define

We construct an obstruction theory for relative Hilbert schemes in the sense of [BF] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V , our obstruction theory determines a virtual fundamental class [[Hilb m V ]] ∈ A m(m−k) 2 (Hilb m V), which we use to define Poincaré invariants These maps are invariant under deformations, satisfy a blow-up formula , and a wall crossing formula for surfaces with pg(V) = 0. In the case q(V) ≥ 1, we calculate the wall crossing formula explicitely in terms of fundamental classes of certain Brill-Noether loci for curves. We determine the invariants completely for ruled surfaces, and rederive from this classical results of Nagata and Lange. The invariant (P