On the First Infinitesimal Neighborhood of a Linear Configuration of Points in P

We consider the following open questions. Fix a Hilbert function h, that occurs for a reduced zero-dimensional subscheme of P. Among all subschemes, X, with Hilbert function h, what are the possible Hilbert functions and graded Betti numbers for the first infinitesimal neighborhood, Z, of X (i.e. the double point scheme supported on X)? Is there a minimum (h) and maximum (h) such function? The numerical information encoded in h translates to a type vector which allows us to find unions of points on lines, called linear configurations, with Hilbert function h. We give necessary and sufficient conditions for the Hilbert function and graded Betti numbers of the first infinitesimal neighborhoods of all such linear configurations to be the same. Even for those h for which the Hilbert functions or graded Betti numbers of the resulting double point schemes are not uniquely determined, we give one (depending only on h) that does occur. We prove the existence of h, in general, and discuss h. Our methods include liaison techniques.