Fat points schemes on a smooth quadric

We study zero-dimensional fat points schemes on a smooth quadric image, and we characterize those schemes which are arithmetically Cohen–Macaulay (aCM for short) as subschemes of Q giving their Hilbert matrix and bigraded Betti numbers. In particular, we can compute the Hilbert matrix and the bigraded Betti numbers for fat points schemes with homogeneous multiplicities and whose support is a complete intersection (CI for short). Moreover, we find a minimal set of generators for schemes of double points whose support is aCM.