On the homogeneous ideal of collinear punctual subschemes of P^2

Fix positive integers $s, n_i, 1\leq i\leq s$, a finite number of lines, $D_1,\cdots,D_s$ of $\textbf{CP}^2$, points $P1,\cdots,Ps$ with $P_i\in D_i$ for all $i$ and let $Z(i)$ be the length $n_i$ subscheme of $D_i$ with support $P_i$. Set $Z:= \cup_{1\leq i\leq s}Z(i)$. Assume $D_i$ and $P_i$ general. Here we show (under mild assumptions on the integers $n_i$) that the homogeneous ideal of $Z$ has the expected number of generators in each degree and hence we compute the minimal free resolution of $Z$.