  We construct the derived version of the Hilbert scheme parametrizing
subschemes in a given projective scheme X with given Hilbert polynomial h. This
is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family
of commutative (up to homotopy) dg-algebras, which over the usual Hilbert
scheme is just given by truncations of the homogeneous coordinate rings of
subschemes in X. In particular, RHilb_h(X) differs from RQuot_h(O_X), the
derived Quot scheme constructed in our previous paper (math.AG/9905174) which
carries only a family of A-infinity modules over the coordinate algebra of X.
  As an application, we construct the derived version of the moduli stack of
stable maps of (variable) algebraic curves to a given projective variety Y,
thus realizing the original suggestion of M. Kontsevich.
