On the homogeneous ideal of collinear punctual subschemes of \(\mathbb{P}^2\) (Q1366592)

Fix positive integers \(s,n_i\), \(1\leq i\leq s\), a finite number of lines, \(D_1,\dots, D_s\), of \(\mathbb{C}\mathbb{P}^2\), points \(P_1,\dots,P_s\) 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 \(Z:= \bigcup_{1\leq i\leq s}Z(i)\). 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 \(Z\). The proof depends heavily on the proofs in a paper by \textit{A. Eastwood} [Manuscr. Math. 67, No.~3, 227-249 (1990; Zbl 0722.41010)], in which the corresponding problem for the postulation was solved using degenerations of collinear schemes to suitable 0-dimensional subschemes of \(\mathbb{C}\mathbb{P}^2\) and an inductive technique called the Horace method [cf. \textit{A. Hirschowitz}, Manuscr. Math. 50, 337-388 (1985; Zbl 0571.14002)].