We determine an upper bound for the cohomological dimension of the complement of a closed subset in a projective variety which possesses an appropriate stratification. We apply the result to several particular cases, including the Bialynicki-Birula stratification; in this latter case, the bound is optimal.
In this paper we obtain a cohomological splitting criterion for locally free sheaves on arithmetically Cohen-Macaulay surfaces with cyclic Picard group, which is similar to Horrocks' splitting criterion for locally free sheaves on projective spaces. We also recover a duality property which identifies a general K 3 surface with a certain moduli space of stable sheaves on it, and obtain examples of stable, arithmetically Cohen-Macaulay, locally free sheaves of rank two on general surfaces of degree at least five in P 3 .
Abstract. We prove that the tensor product of two line bundles, one being q-ample and the other with sufficiently low-dimensional base locus, is still q-ample.
The resultThe goal of this note is to prove the following property of the q-ample cone of a projective variety.Theorem A. Let X be a normal, irreducible projective variety defined over an algebraically closed field of characteristic zero. Consider A, L ∈ Pic(X) and denote the stable base locus of A by sb(A). We assume that L is q-ample andThe classes of the q-ample line bundles form an open cone in the vector spacegenerated by invertible sheaves (line bundles) on X modulo numerical equivalence (cf. [1,3]). The tensor product of two q-ample line bundles is not q-ample in general (cf. [8, Theorem 8.3]), and therefore the q-ample cone is, usually, not convex. This situation contrasts the classical case of ample line bundles, corresponding to q = 0, which generate a convex cone. Actually, it is well-known that the ample cone of a projective variety is stable under the addition of a numerically effective (nef) term.Moreover, Sommese proved in [7, Corollary 1.10.2] that the tensor product of two globally generated, q-ample line bundles is still q-ample. However, the concept of q-ampleness used in loc. cit. is defined geometrically and it is based on the global generation of the line bundles.For this reason, it is natural to ask whether the q-ample cone is stable under the addition of suitable terms; by abuse of language, we call such a feature a 'convexity property'. The theorem stated above can be viewed as an answer to this question. 1991 Mathematics Subject Classification. 14F17, 14C20.
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