On some properties of partial intersection schemes

Partial intersection subschemes of $\mathbb{P}^r$ of codimension c were used to furnish various graded Betti numbers which agree with a fixed Hilbert function. Here we study some further properties of such schemes; in particular, we show that they are not in general licci and we give a large class of them which are licci. Moreover, we show that all partial intersections are glicci. We also show that for partial intersections the first and the last Betti numbers, say m and p respectively, give bounds each other; in particular, in the codimension 3 case we see that $\lceil\frac{p+5}{2}\rceil\leq m\leq 2 p+1$ and each m and p satisfying the above inequality can be realized.