On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb P^2$

We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. 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 ($h^{\min}$) and maximum ($h^{\max}$) such function? The numerical information encoded in $h$ translates to a {\it type vector}, which allows us to find unions of points on lines, called {\it 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 {\it 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 $h^{\max}$, in general, and discuss $h^{\min}$. Our methods include liaison techniques.