By D. E. Lerner, P. D. Sommers

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The second method (due to Horrocks and Barth) will not be pursued further here [see Christ's lecture in these proceedings and reference 1]. The idea of the first method is as follows. vector bundle over P3 • E8 Then there exists a positive integer q such that L(q) has a section which vanishes along a curve therefore, E8 line bundle. A Let E be a 2-dimensional r in P3 • On P 3 - L(q) is an extension of the trivial line bundle by another After tensoring with L(-q) we arrive at the result that restricted to P3 - r, is an extension of L(-q) by L(q).

We call Y a cocycle if cSy Define the H~U } (X,~) i Note: p H{U } (X, I&), i th E== cohomology = 0; Then we have we call Y a coboundary if Y = ~by: (additive group)/(additive group ) of p-cocycles of p-coboundaries • as defined, depends on the covering {Ui}.

Nger [2] using methods of differential geometry). However, it is not yet known (except fork= 1, 2) whether M'(k) is connected. There are two principal methods for studying rank 2 vector bundles on P 3 • one is via curves, the other via monads. The curve method goes back to Serre (1960), was used by Horrocks (1968), and was apparently rediscovered independently by Barth and Van de Ven (1974) and Grauert and Mulich (1975). idea is this. It is the main technique used in [8). The Tensor the bundle E by a multiple of the Hopf line bundle 0(1) so that the new bundle E(n) has plenty of global sections.