Zero-dimensional subschemes of projective spaces related to double points of linear subspaces and to fattening directions

Fix a linear subspace $V\subseteq \mathbb {P}^n$ and a linearly independent set $S\subset V$. Let $Z_{S,V} \subset V$ or $Z_{s,r}$ with $r:= \dim (V)$ and $s=\sharp (S)$, be the zero-dimensional subscheme of $V$ union of all double points $2p$, $p\in S$, of $V$ (not of $\mathbb {P}^n$ if $n>r$). We study the Hilbert function of $Z_{S,V}$ and of general unions in $\mathbb {P}^n$ of these schemes. In characteristic $0$ we determine the Hilbert function of general unions of $Z_{2,1}$ (easy), of $Z_{2,2}$ and, if $n=3$, general unions of schemes $Z_{3,2}$ and $Z_{2,2}