Non degenerate projective curves with very degenerate hyperplane section

We study the Hilbert scheme of non degenerate locally Cohen-Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component H(n,d,g) of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of H(n,d,g).