  Simon Donaldson and Ivan Smith recently studied symplectic surfaces in
symplectic 4-manifolds X by introducing an invariant DS associated to any
Lefschetz fibration on blowups of X which counts holomorphic sections of a
relative Hilbert scheme that is constructed from the fibration. Smith has shown
that DS satisfies a duality relation identical to that satisfied by the Gromov
invariant Gr introduced by Clifford Taubes, which led Smith to conjecture that
DS=Gr provided that the fibration has high enough degree. This paper proves
that conjecture. The crucial technical ingredient is an argument which allows
us to work with curves C in the blown-up 4-manifold that are made holomorphic
by an almost complex structure which is integrable near C and with respect to
which the fibration is a pseudoholomorphic map.
