Math 8174 (Fall 2017)
Algebraic curves and Picard groups

Goals of the course

We will develop the necessary foundations from algebraic geometry to state and prove Deligne's "universal coefficients theorem" for algebraic curves. The theorem states that there is a universal cohomology theory — the Picard group — for curves satisfying some rough algebraic analogues of excision and homotopy invariance. In order to see the universality of the Picard group, we must consider it not just as a group but as an abelian variety.

One consequence of the universal coefficients theorem is a classification of abelian covers of algebraic curves. This is known as "geometric class field theory" because it is analogous to the classification of abelian extensions of number fields in class field theory.

The course will use the language of schemes, which will be assumed. It is possible to get something out of this course without prior experience with schemes, especially in the first component on sheaf cohomology. Later parts of the course will use schemes and commutative algebra more heavily. Category theory will be used throughout.

Here are the main topics I plan to cover:

  1. Sheaf cohomology (Hartshorne, Ch. 3)
  2. Algebraic curves (Hartshorne, Ch. 4; Mumford)
  3. Construction of the Picard scheme (Grothendieck; Kleiman; Serre)
  4. Universal coefficients (Deligne; Serre)


Classroom: ECCR 116
Meeting times: MWF, 12 – 12:50
Office: Math 204
Office hours: calendar


The following are the references I'll be consulting; I will probably add to this list as the semester goes on. I will assign readings from some of them, but not necessarily all. All are available online, either from the arXiv or from Springer Link. Note that some are in French.


  1. The Picard group
  2. Stacks and Grothendieck topologies
  3. Curves
  4. October 6: coherent cohomology
  5. September 29: Čech cohomology
  6. September 11: sheaf cohomology
  7. September 6: derived functors and sheaf cohomology
  8. September 1: abelian categories, sheaves of abelian groups form one
  9. August 28: overview of the course, review of sheaves


I will sometimes give problems in class or as homework. You do not have to submit these, but I expect you to demonstrate an honest effort towards solving them, either by submitting them or discussing them with me.