Some very ample and base point free linear systems on generic rational surfaces. (English) Zbl 1054.14011

Chauvin, Stéphane; De Volder, Cindy

Math. Nachr. 245, 45-66 (2002).

From the introduction: The study of linear systems on general blowing-ups of the projective plane has received considerable attention. When the number of points blown up is at most nine, the situation is well understood [B. Harbourne, Math. Ann. 272, 139–153 (1985; Zbl 0545.14003)], but, for ten or more points, the basic questions of the dimension, base point freeness and very ampleness remain unillucidated. Here we are concerned with base point freeness and very ampleness and we prove that Alexander’s conjecture [J. Alexander, Generalised Harbourne-Hirschowitz conjecture, Preprint, http://arxiv.org/abs/math.AG/0111188] holds in a special, but nontrivial, case. This case encompasses two known cases, the first, which we just mentioned, is for at most nine points and the second is a result of J. D’Almeida and A. Hirschowitz [Math. Z. 211, 479–483 (1992; Zbl 0759.14004)] which considers linear systems coming from a smooth base locus.

MSC:

14C20	Divisors, linear systems, invertible sheaves
14J26	Rational and ruled surfaces
14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F17	Vanishing theorems in algebraic geometry

Keywords:

general blowing-up of the projective plane; Alexander’s conjecture

Citations:

Zbl 0545.14003; Zbl 0759.14004