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Generically ample divisors on normal Gorenstein surfaces. (English) Zbl 0687.14005

Singularities, Proc. IMA Participating Inst. Conf., Iowa City/Iowa 1986, Contemp. Math. 90, 1-19 (1989).
[For the entire collection see Zbl 0668.00006.]
Let (S,L) be a pair consisting of a nef and big line bundle L on an irreducible normal Gorenstein surface. The aim of this paper is to study the adjunction mapping of such a pair. Roughly speaking, if \(K_ S\) is a canonical divisor of S then - modulo certain minimality assumptions - one proves that if there is an \(n>0\) such that \(H^ 0(S,L^ n\otimes K^ n_ S)\neq 0\) then for \(m>>0\) the line bundle \(L^ m\otimes K^ m_ S\) is spanned.
If \(f:\quad S\to P\) is the morphism associated to \(| L^ m\otimes K^ m_ S|\) into a projective space, then one has three possibilities according to the possible value for dim(f(S)). In each of these cases the result gives explicit information.
Note that the case when L is ample had been studied by the second named author [see A. Sommese, Abh. Math. Semin. Univ. Hamb. 55, 151-160 (1985; Zbl 0609.14003)]. The motivation of the result in the present generality lies in the study of the pairs (S,L) such that S is the minimal desingularization \(f:\quad S\to S'\subset P\) of an irreducible surface S’ of a projective space P and \(L=f^*({\mathcal O}_ P(1))\).
Reviewer: L.Bădescu

MSC:

14C20 Divisors, linear systems, invertible sheaves
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14J25 Special surfaces