  In this paper, we discuss some necessary and sufficient condition for a curve
to be arithmetically Cohen-Macaulay, in terms of its general hyperplane
section. We obtain a characterization of the degree matrices that can occur for
points in the plane that are the general hyperplane section of a non
arithmetically Cohen-Macaulay curve of P^3. We prove that almost all the degree
matrices with positive subdiagonal that occur for the general plane section of
a non arithmetically Cohen-Macaulay curve of P^3, arise also as degree matrices
of the general plane section of some smooth, integral, non arithmetically
Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary
condition on the graded Betti numbers of the general hyperplane section of an
arithmetically Buchsbaum, (non arithmetically Cohen-Macaulay) curve in P^n. For
curves in P^3, we show that any set of Betti numbers that satisfies that
condition, can be realised as the Betti numbers of the general plane section of
an arithmetically Buchsbaum, non arithmetically Cohen-Macaulay curve. We also
show that the matrices that arise as degree matrix of the general plane section
of an arithmetically Buchsbaum, integral, (smooth) non arithmetically
Cohen-Macaulay space curve are exactly those that arise as degree matrix of the
general plane section of an arithmetically Buchsbaum, non arithmetically
Cohen-Macaulay space curve and have positive subdiagonal. We also prove some
bounds on the dimension of the deficiency module of an arithmetically Buchsbaum
space curve, in terms of the degree matrix of the general plane section of the
curve.
