  Let F be a smooth surface in a smooth projective threefold T, and let X=2F be
the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve
C in X gives rise to two effective divisors on F, namely the curve part P of
the intersection of C and F, and the curve R residual in C to this
intersection. We show that a general deformation of R on F lifts to a
deformation of C on X when a certain cohomology group vanishes. In our paper
"Hilbert Schemes of Degree Four Curves" we use this result to prove the
connectedness of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space
curves of degree four and arbitrary arithmetic genus g.
