Nodal curves and postulation of generic fat points on \(\mathbb{P}^2\)
type: Item
instance of: scholarly article
title: Nodal curves and postulation of generic fat points on \(\mathbb{P}^2\)
zbMATH Open document ID: 0947.14015
author: E. Ballico
author: Luca Chiantini
published in: Archiv der Mathematik
publication date: 2000-11-05T00:00:00Z
review text: 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)].
reviewed by: Ana J. Reguera-López
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 1292441
zbMATH Keywords: irreducible plane curves
zbMATH Keywords: fat point
zbMATH Keywords: node
zbMATH Keywords: prescribed singularities
MaRDI profile type: Publication
full work available at URL: s000130050295
OpenAlex ID: W2062321215
DOI: 10.1007/S000130050295
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description: scientific article; zbMATH DE number 1292441