Nodal curves and postulation of generic fat points on \(\mathbb{P}^2\)
The paper deals with the problem of the existence of irreducible plane curves of given degree with prescribed singularities, more precisely, the prescribed singularities are all fat points and most of them nodes. A sufficient condition is given, which consists of linear inequalities, on a set of positive integers \(t, r, e, d_1,\dots{},d_r\) in order that, for a general multi-jet \[ Z= \Bigl( \bigcup_{1 \leq j \leq r} d_j P_j\Bigr) \cup \Bigl( \bigcup_{r \leq j \leq e} 2 P_j\Bigr) \] (i.e. general points \(P_1,\dots{},P_r, P_{r+1},\dots{}, P_e\) with multiplicities \(d_1,\dots{},d_r, 2,\dots{},2\)) we have \(h^1({\mathbb{P}}^2, {\mathcal I}_Z(t))=0\) where \({\mathcal I}_Z\) is the ideal sheaf of \(Z\). The main idea for the proof is the Horace lemma appearing in a paper by \textit{A. Hirschowitz} [J. Reine Angew. Math. 397, 208-213 (1989; Zbl 0686.14013)].
scientific article; zbMATH DE number 1292441