A basic problem in algebraic geometry is the classification of
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