## Graduate Seminar on Advanced Algebra (S4A3) - The Picard Functor - Sommersemester 2015

Prof. Dr.
Michael Rapoport

Dr. Eugen Hellmann

Kontakt: hellmann (ergänze @math.uni-bonn.de)

Classically, the Picard group Pic(X) of a manifold or an algebraic variety X is the set of isomorphism classes of line bundles on X with group law given by the tensor product. Equivalently the Picard group can be defined as the first cohomology H^{1}(X, G_{m}) of the multiplicative group.

In algebraic geometry one is often interested in the variation of objects in families and in particular in the existence of a universal family. In the case of line bundles we can fix a variety X over an (algebraically closed) field k and view a lines bundle on the product X x T
of X with a k-scheme T as a family of line bundles parametrized by the points of T.
More generally, given a base scheme S and an S-scheme X we can consider the functor Pic_{X} that assigns to an S-scheme T the set of isomorphism classes of line bundles on the fiber product X x_{S} T.

It turns out that this functor can not be representable as there is an action of Pic(T) on Pic_{X}(T) given by tensorizing a line bundle on the fiber product X x_{S} T with the pullback of a line bundle on T. Hence we will define the relative Picard functor Pic_{X/S} as the quotient of Pic_{X} by this action.

In this seminar we want to study the Picard functor (respectively its sheafification for a suitable topology) and some of its variants and prove a general representability theorem following Grothendieck's approach.

## Prerequisites

We assume familiarity with the basic concepts of Algebraic Geometry, roughly in the amount of chapters II and III of Hartshorne's book.

## Time and Place

Tuesday 16-18h, MZ Room 0.006

## Program

The detailed program can be found here .

## Organizational meeting

Tuesday 03.02.2015 um 16h (c.t.), MZ Room 1.007.If you can not come to the organizational meeting please e-mail in advance.

## References

- B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure:
*Fundamental Algebraic Geometry*, Math. Surveys and Monographs 123, American Math. Soc.

*Last modified: 01. 02. 2015, Eugen Hellmann*