Regularity of linear systems of plane curves
type: Item
instance of: scholarly article
title: Regularity of linear systems of plane curves
zbMATH Open document ID: 0702.14003
DOI: 10.1016/0021-8693(89)90142-7
author: Alessandro Gimigliano
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathbb{P}}_ k^ 2\) be the projective plane, k an algebraically closed field, and let Z be the subscheme of \({\mathbb{P}}_ k^ 2\) which has support on the points \(P_ 1,...,P_ s\) with multiplicity \(m_ i\) at \(P_ i\), \(i=1,...,s\). - Consider the linear system of plane curves having Z as scheme of base points and let \(I_ t\) be the associated subspace of the vector space \(R_ t\) of all forms of degree \(t\) in the homogeneous coordinates \([x_ 0,x_ 1,x_ 2]\) on \({\mathbb{P}}_ k^ 2\). The linear system \(I_ t\) is said to be regular if \(\dim (I_ t)=\dim (R_ t)-\sum^{s}_{i=1}(m_ i(m_ i-1))/2.\) Define \(\tau (Z)=\min (t\), \(I_ t\) regular). The bound \(\tau\) (Z)\(\leq \sum^{s}_{i=1}m_ i -1 \) is classically known and a better bound was given by \textit{B. Segre} [Atti Convegno Intern. Geom. Algebrica, Torino 1961, 15-33 (1962; Zbl 0104.389] under additional condition on Z. \textit{B. Harbourne} [Trans. Am. Math. Soc. 289, 213-226 (1985; Zbl 0609.14004) and Algebraic Geometry, Proc. Conf., Vancouver/B.C. 1984, CMS Conf. Proc. 6, 95-111 (1986; Zbl 0611.14002)] has given an algorithm to find out if a linear system is regular in the case of points contained in a curve of degree \(\leq 3.\) The aim of the paper under review is to generalize some of the results by Harbourne to sets of generic points, and the main result is as follows: Let \(S=\{P_ 1,...,P_ s\}\) be contained in the nonsingular locus of an irreducible and reduced plane curve of degree \(d\geq 3\) and suppose that \(\tau\) (S)\(\geq d\). Then, if \(m_ 1\geq...\geq m_ s\geq 0\), \(\tau\) (Z)\(\leq \sum^{s}_{i=1}m_ i\) unless \(s=d(d+3)/2\) and \(m_ 1=...=m_ s=2\) in which case \(\tau\) (Z)\(\leq \sum^{d}_{i=1}m_ i +1.\) The case \(\tau (Z)=\sum^{d}_{i=1}m_ i +1 \) is analyzed in some detail. If S is a generic set and \(s\leq d(d+3)/2\) it is proved that \(\tau\) (Z)\(\leq \sum^{d}_{i=1}m_ i \). The method used in studying the problem consists essentially in ``moving'' from \({\mathbb{P}}_ k^ 2…
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 4150364
zbMATH Keywords: regularity of linear systems of plane curves
reviewed by: A. Del Centina
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90142-7
OpenAlex ID: W2010795701
cites work: Q3887589
cites work: Q3729039
cites work: Q3795781
cites work: Lectures on Riemann Surfaces: Jacobi Varieties
cites work: Q3313974
cites work: Complete Linear Systems on Rational Surfaces
cites work: Q3751693
cites work: La methode d'Horace pour l'interpolation à plusieurs variables
cites work: Q3929878
cites work: Elliptic curves. Notes from postgraduate lectures given in Lausanne 1971/72
cites work: Sui teoremi di Bezout, Jacobi e Reiss
cites work: Q3289515
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description: scientific article; zbMATH DE number 4150364