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By C. R. F. Maunder

Thorough, smooth therapy, basically from a homotopy theoretic point of view. subject matters comprise homotopy and simplicial complexes, the basic workforce, homology conception, homotopy conception, homotopy teams and CW-Complexes and different subject matters. each one bankruptcy comprises routines and proposals for additional examining. 1980 corrected variation.

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The map H°(Is(n + 1)) -> H°(IH(n + 1)) is not surjective). Proof. Consider the cohomology of the natural exact sequences 0 —-> Os(n) —> Os(n + 1) —• OH{n + 1) —• 0 resp. 0 —• Is(n) —> Is(n + 1) —> IH(n + 1) —• 0 when H varies. 9) Lemma. Let S be a smooth surface inP4. then hl(Os(2)) = 0. If H • K < d - 4 and ^(OsW) < 2, Proof. Assume that h1{Os{2)) > 0. Now, H • (H + K) < 2d so hl{OH(2)) = 0 for any integral H. Thus hl(Os(2)) < 2. 8) and its proof the variety V C P 4 parametrizing hyperplane sections for which h1{On{2)) > 0 contains a plane, so there is a line I c P 4 contained in a net of hyperplanes for which hl{Ou{2)) > 0.

38 AURE & RANESTAD: The smooth surfaces of degree 9 in P4 Since /i 1 (Os(l)) = 4 — x, by Seven, and OH(2H) is nonspecial for a smooth H, h\Os{n)) 2. Hence = X + 30 + h\Os&)) < 34 and 5 is contained in a cubic hypersurface. Similarly /i°(0s(4)) < 62, so S is contained in a quartic which is not a multiple of the cubic. Therefore 5 is linked (3,4) to a cubic surface (possibly singular/reducible). 9), the hyperplane section of 5 is linked to a cubic curve of genus 0. By linkage, this curve is locally Cohen-Macaulay.

R m Corollary 6. Let E be a semistable vector bundle which is generically generated by global sections and assume that h°(E (8> u>c) ^ 0. Then Proof. Since E (g) UJQ satisfies the hypothesis of corollary 5, 2 ) +R + d+R _ Rg = *+ R m We have thus shown that h°(E) < | + R whenever two of the following three conditions are satisfied: 1) E is semistable. 2) E is generically generated by global sections. 3) E (8) uc is generically generated by global sections. We thus Conjecture. Let E be a semistable vector bundle of rank R and degree d on a smooth connected curve C.

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