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A PRYM CONSTRUCTION FOR THE COHOMOLOGY OF A CUBIC HYPERSURFACE

Published online by Cambridge University Press:  01 November 1999

E. IZADI
E. IZADI
Affiliation:
Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, GA 30602-7403, U.S.A., email:[email protected]
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Abstract

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over ${\Bbb P}^2$ and the Prym variety of a naturally defined étale double cover of the discriminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold, and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over ${\Bbb P}^2$. We further generalize the isomorphism to the primitive {\em cohomology} of a smooth cubic hypersurface in ${\Bbb P}^n$. We give two applications of our construction: one is a special case of the generalized Hodge conjectures and the other is an Abel-Jacobi isomorphism.

1991 Mathematics Subject Classification: primary 14J70; secondary 14J45.

Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 79 , Issue 3 , November 1999 , pp. 535 - 568
Copyright
1999 London Mathematical Society

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