Vector bundles of rank 2 and linear systems on algebraic surfaces. (English) Zbl 0663.14010

Reider, Igor

Ann. Math. (2) 127, No. 2, 309-316 (1988).

Let X be a smooth complex algebraic surface, L a nef divisor on X and \(K_ X\) the canonical divisor on X. In his main theorem the author constructs an effective divisor E depending on \(L^ 2\) and going through a base point of \(| K_ X+L|\) or pairs of points not separated by \(| K_ X+L|\) and gives all possible values of L.E and \(E^ 2\). He uses the notion of a divisor being in special position with respect to \(| K_ X+L|\) to get a decomposition of \(L=M+E\) due to Bogomolov [M. Reid, Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 623-642 (1977; Zbl 0478.14003)].
Then he shows that his theorem implies:
(a) results of Bombieri on the pluricanonical maps on surfaces of general type,
(b) some results on surfaces with Kodaira dimension 0, due to Beauville, and
(c) Sommese’s and Van de Ven’s results on the adjunction mapping.

Reviewer: A.Papantonopoulou

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14J10	Families, moduli, classification: algebraic theory
14C20	Divisors, linear systems, invertible sheaves

Keywords:

nef divisor; divisor being in special position; pluricanonical maps; Kodaira dimension 0; adjunction mapping

Citations:

Zbl 0478.14003