When the positivity of the h-vector implies the Cohen-Macaulay property

We study relations between the Cohen-Macaulay property and the positivity of the h-vector of a locally Cohen-Macaulay equidimensional closed subscheme X of the projective space of dimension n over a field K, showing that these two conditions are equivalent for those X which are close to a complete intersection Y (of the same codimension) in terms of the difference between the degrees. More precisely, let X be contained in Y , either of codimension two with deg(Y )- deg(X) <= 5 or of codimension 3 with deg(Y ) - deg(X) <= 3. Over a field K of characteristic 0, we prove that X is arithmetically Cohen-Macaulay if and only if its h-vector is positive, improving results of a previous work. If X is a curve, this result holds in every characteristic different from 2. We find also other classes of schemes for which the positivity of the h-vector implies the Cohen-Macaulay property and provide several examples.