In this paper we apply the Beilinson theorem [Functional Anal. Appl. 12 (1978), 214-216] to the following problems. (1) We give sufficient cohomological conditions in order that a coherent sheaf on IP n or on the quadric contains äs direct summand a generator of the derived category (i. e. the line bundles, the bundles ofp-forms on P", the spinor bundles and the bundles \p t introduced by Kapranov [Inv. Math. 92 (1988), 479-508].(2) We characterize the indecomposable sheaves of order one (with respect to H 1 and H 2 )onP 3 and we show that also the diameter is one. (3) We give a new proof of the key theorem which Chang uses to characterize the arithmetically Buchsbaum subschemes of codimension 2 in P".
The aim of this paper is to describe the structure of Fano bundles in dimension > 4. Introduction. In this paper rank 2 vector bundles E on projective spaces Ψ n and quadrics Q n are investigated which enjoy the additional property that their projectized bundles Ψ(E) are Fano manifolds, i.e. have negative canonical bundles. Such bundles are shortly called Fano bundles. Up to dimension 3 Fano bundles are completely classified by [SW], [SW], [SW"], [SSW]. The aim of this paper is to describe the structure of Fano bundles in dimension > 4. Namely we prove the following MAIN THEOREM. Let E be a rank 2 Fano bundle on Ψ n or Q n , n > 4. Then up to some explicit exceptions on Q 4 and Q$ (see ex. (2.1), (2.2), (2.3)), E splits into a direct sum of line bundles. A rank 2 bundle E on Ψ n is Fano if and only if the "Q-vector bundle" E (det£*)/2 ® &{?ψ) is ample, i.e.
Abstract.We prove that the special instanton bundles of rank 2« on P2n+1(C) with a symplectic structure studied by Spindler and Trautmann are stable in the sense of Mumford-Takemoto. This implies that the generic special instanton bundle is stable. Moreover all instanton bundles on P5 are stable. We get also the stability of other related vector bundles.
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