  We discuss the minimal free resolution of an irreducible projective subscheme
X. If X is also reduced, we focus on the case when its degree equals two plus
the codimension. The set of all possible graded Betti numbers is described if
the codimension is two. In general, there are only partial results. Then we
discuss Cohen-Macaulay structures on a linear subspace. For such curves of
degree two, the classification has been obtained by Notari, Spreafico, and the
author. In general, the classification problem seems rather difficult. Vatne
has shown that such a classification of structures whose degree is at most
three, is equivalent to Hartshorne's conjecture on smooth varieties of
codimension two.
