Postulation of subschemes of irreducible curves on a quadric surface (Q1567576)

Consider a 0-dimensional scheme \(X\) which is contained in an irreducible curve \(C\) in \(\mathbb{P}^3\), which in turn is contained in a smooth quadric surface \(Q\). Which are the possibilities for the Hilbert function (postulation) of such an \(X\)? The paper gives an answer (i.e a complete characterization of all possible Hilbert functions for which it exists \(X\) as above) when \(C\) is ``good enough'', in particular when \(C\) is a complete intersection, arithmetically Cohen-Macaulay, or arithmetically Buchsbaum. In the general case conditions on the Hilbert function are given.