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Comparison theorems for conic bundles

Published online by Cambridge University Press:  24 October 2008

P. E. Newstead
P. E. Newstead
Affiliation:
University of Liverpool
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Extract

In (10), M. S. Narasimhan and S. Ramanan proved a theorem to the effect that a certain conic bundle associated with a non-singular quadratic complex does not come from a vector bundle ((10), proposition 8·1); a similar topological result was proved in (12). In the course of attempting to extend these results to the singular case, I found that I wanted to use some results on conic bundles which were not readily available in the literature. The object of this note is to give proofs of these results; the work on quadratic complexes is still in progress and the first part will appear shortly (13). A further application will appear in (14).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 1 , July 1981 , pp. 21 - 31
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Artin, M., Grothendieck, A. and Verdier, J. L.Théorie des topos et cohomologie étale des schémas (SGA4), tome 3. Lecture Notes in Mathematics, no. 305 (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
(2)Artin, M. and Mumford, D.Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. 25 (1972), 75–95.CrossRefGoogle Scholar
(3)Clemens, C. H. and Griffiths, P. A.The intermediate Jacobian of the cubic threefold. Ann. of Math. 95 (1972), 281–356.CrossRefGoogle Scholar
(4)Giraud, J.Cohomologie non abélienne. Grundlehren Bd. 179 (Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
(5)Grothendieck, A.Sur quelques points d'algèbre homologique. Tôhoku Math. J. 9 (1957), 119–221.Google Scholar
(6)Grothendieck, A. Le groupe de Brauer I, II, III. In Dix exposés sur la cohomologie des schémas (North-Holland, Amsterdam, 1968), pp. 46–188.Google Scholar
(7)Grothendieck, A. and Dieudonné, J.Éléments de géométrie algébrique, chapitre III, seconde partie Inst. Hautes Études Sci. Publ. Math. no. 17 (1963).Google Scholar
(8)Hartshorne, R.Algebraic geometry. Graduate Texts in Mathematics, vol. 52 (Springer-Verlag, New York, Heidelberg, Berlin, 1977).Google Scholar
(9)Narasimhan, M. S. and Ramanan, S.Moduli of vector bundles on a compact Riemann surface. Ann. of Math. 89 (1969), 14–51.CrossRefGoogle Scholar
(10)Narasimhan, M. S. and Ramanan, S. Vector bundles on curves. In Algebraic Geometry (papers presented at the Bombay Colloquium 1968) (O.U.P., India 1969), pp. 335–346.Google Scholar
(11)Newstead, P. E.Stable bundles of rank 2 and odd degree over a curve of genus 2. Topology 7 (1968), 205–215.CrossRefGoogle Scholar
(12)Newstead, P. E.A note on quadratic complexes. J. London Math. Soc. 5 (1972), 748–752.CrossRefGoogle Scholar
(13)Newstead, P. E. Quadratic complexes II. (To appear.)Google Scholar
(14)Newstead, P. E. On the homology and the Picard group of a moduli space of bundles on ℙ3. (To appear.)Google Scholar
(15)Reid, M. A. The complete intersection of two or more quadrics (Thesis, Cambridge University, 1972).Google Scholar
(16)Serre, J. P. Espaces fibrés algébriques. Séminaire Chevalley (1958).Google Scholar
(17)Spanier, E. H.The homology of Kummer manifolds. Proc. Amer. Math. Soc. 7 (1956), 155–160.CrossRefGoogle Scholar
(18)Tjurin, A. N.Five lectures on three-dimensional varieties. Uspekhi Mat. Nauk 27, no. 5 (1972), 3–50; Russian Math. Surveys 27, no. 5 (1972), 1–53.Google Scholar