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

Prof. Dr. Michael Rapoport
Dr. Eugen Hellmann
Kontakt: hellmann (ergänze

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, Gm) 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 PicX that assigns to an S-scheme T the set of isomorphism classes of line bundles on the fiber product X xS T.

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


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


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.


  • 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