  We construct an obstruction theory for relative Hilbert schemes in the sense
of Behrend-Fantechi 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 $[[
\Hilb^m_V ]]$, which we use to define Poincare invariants
  (P^+_V,P^-_V): H^2(V,Z) --> \Lambda^* H^1(V,Z) x \Lambda^* H^1(V,Z).
  These maps are invariant under deformations, satisfy a blow-up formula, and a
wall crossing formula for surfaces with $p_g(V)=0$. We determine the invariants
completely for ruled surfaces, and rederive from this classical results of
Nagata and Lange. The invariant $(P^+_V,P^-_V)$ of an elliptic fibration is
computed in terms of its multiple fibers.
  We conjecture that our Poincare invariants coincide with the full
Seiberg-Witten invariants of Okonek-Teleman computed with respect to the
canonical orientation data. The main evidence for this conjecture is based on
the existence of an Kobayashi-Hitchin isomorphism which identifies the moduli
spaces of monopoles with the corresponding Hilbert schemes. We expect that this
isomorphism identifies also the corresponding virtual fundamental classes. This
more conceptual conjecture is true in the smooth case.
