Mathematical Institute of the University of Bonn

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

Prof. Dr. Michael Rapoport
Dr. Eugen Hellmann
Contact: hellmann (add @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 H¹(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.