Regularity of linear systems of plane curves
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 \({\mathbb{P}}_ k^ 2\) to the blowing-up X of \({\mathbb{P}}_ k^ 2\) at \(P_ 1,...,P_ s\) and in calculating the cohomology of a given divisor D on X by the so called ``Horace method'' [\textit{A. Hirschowitz}, Manuscr. Math. 50, 337-388 (1985; Zbl 0571.14002)] consisting in splitting D as \(D'+D''\) one of which has in its class a reduced and irreducible curve, and the other one has known cohomology.
regularity of linear systems of plane curves
scientific article; zbMATH DE number 4150364
scientific article