A basic problem in algebraic geometry is the classification of projective varieties. A very interesting case is when the codimensions of projective varieties are small. One of the motivations to study such varieties is Hartshorne's conjecture. In 1974, he suggested that a nonsingular subvariety of projective space should be a complete intersection, when its codimension is small as compared with the dimension of the projective space. This conjecture is closely related to the question of whether there are vector bundles of small rank on the projective space, which are not direct sum of line bundles. So it is an important task to find such bundles. However this seems a very hard problem. Indeed, there is no such bundle known to exist on the projective space whose dimension is greater than or equal to 6. So a question is: ``How can we attack this problem?''

There are several construction methods known. One of the powerful methods for constructing bundles is Kumar's construction. This construction is based on the well-known Serre's conjecture, which was solved by Quillen and Suslin independently. For a given vector bundle, the theorem of Quillen and Suslin guarantees us the existence of sections that generate the vector bundle on the complement of a hyperplane. The pair of the vector bundle and these sections corresponds to a vector bundle on this hyperplane. Kumar gave necessary and sufficient conditions for a vector bundle on a hyperplane of projective space to be obtained from a vector bundle on the projective space in this way. It turns out that there exists a correspondence between vector bundles on the n-dimensional projective space and vector bundles on the (n-1)-dimensional projective space satisfying certain conditions. By using this correspondence, Kumar established the existence of many rank two vector bundles on projective fourspace in positive characteristics.

A natural question is: ``How can we compute a vector bundle on the n-dimensional projective space from the corresponding bundle on the (n-1)-dimensional projective space? The purpose of this talk is to develop a constructive method for this computation and to show by means of a couple of examples how this method works. This is joint work with Chris Peterson.

Last Updated: March 10 2004 by Hirotachi Abo