Vector bundles on the cubic threefold

Vector bundles on the cubic threefold Arnaud BEAUVILLE To Herb Introduction Let X be a smooth cubic hypersurface in P4 . In their seminal paper [C-G], Clemens and Griffiths showed that the intermediate Jacobian J(X) , an abelian vari- ety defined analytically through Hodge theory, is a fundamental tool to understand the geometry of X . They studied the Fano surface F of lines contained in X , proving that the Abel-Jacobi map embeds F into J(X) and induces an isomor- phism Alb(F) ??? J(X) . They were able to deduce from this the Torelli theorem and the non-rationality of X (a problem which had resisted the efforts of the Italian geometers)1. Mumford noticed that one can express J(X) as a Prym variety and thus give an alternate proof for the non-rationality of X ([C-G], Appendix C); the other results of [C-G] can also be obtained via this approach [B2]. Later Clemens observed that one could use the twisted cubics as well, giving an elegant parametrization of the theta divisor (see (4.2) below). At this point the cubic threefold could be considered as well understood, and the emphasis shifted to other Fano threefolds.

• semi-stable sheaves


• abel-jacobi map embeds


• smooth


• using chern


• map ?


• vector bundle


• p5 -bundle over


• curve ? ?


• stable rank