When the positivity of the \(h\)-vector implies the Cohen-Macaulay property (Q480417)

If \(X \subset \mathbb P^N\) is a closed subscheme of dimension \(d\) with Hilbert function \(H_X (t)\), the \textit{\(h\)-vector} of \(X\) is the sequence \((\Delta^{d+1} H_X (0), \Delta^{d+1} H_X (1), \dots \Delta^{d+1} H_X (s))\), where \(s\) is the largest integer making the last term non-zero. It is known that if \(X\) is arithmetically Cohen-Macaulay (ACM), then its \(h\)-vector is positive (\(h_i > 0\) for \(0 \leq i \leq s\)). The authors give some conditions under which the converse holds for locally Cohen-Macaulay subschemes \(X\). For example, if \(X \subset \mathbb P^3\) is a curve contained in a complete intersection curve \(Y\) with \(\deg Y - \deg X \leq 5\), they prove that \(X\) is ACM if and only if its \(h\)-vector is positive; for curves in higher dimensional projective spaces they prove the same assuming \(\deg Y - \deg X \leq 3\). These results follow from their earlier work [Collect. Math. 62, No. 2, 173--186 (2011; Zbl 1221.14054)]. For \(X \subset \mathbb P^n\) (\(n > 3\)) of codimension 2 contained in a complete intersection \(Y\) with \(\dim Y = \dim X\) and \(\deg Y - \deg X \leq 5\) they again prove that \(X\) is ACM if and only if the \(h\)-vector is positive; for \(X\) of codimension at least three and \(\deg Y - \deg X \leq 3\) they prove the same. The proofs amount to looking at the residual scheme \(Z\) to \(X\) in \(Y\); since \(Z\) has small degree there are just a few possibilities for the first difference of the Hilbert function and these are analyzed to deduce the results.