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A Prym Construction for the Cohomology of a Cubic Hypersurface
Author(s) -
Izadi E.
Publication year - 1999
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611599012046
Subject(s) - mathematics , hypersurface , isomorphism (crystallography) , quadric , pure mathematics , cohomology , conic section , vector bundle , algebra over a field , geometry , chemistry , crystal structure , crystallography
Mumford defined a natural isomorphism between the intermediate jacobian of a conic‐bundle over 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 P 2 . We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in 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.