Minimal free resolutions of projective subschemes of small degree

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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.