# 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:

- Sheaf cohomology (Hartshorne, Ch. 3)
- Algebraic curves (Hartshorne, Ch. 4; Mumford)
- Serre duality on curves (Altman & Kleiman, Ch. 5; Tate)
- Construction of the Picard scheme (Grothendieck; Kleiman; Serre)
- Universal coefficients (Deligne; Serre)

## Logistics

Classroom: ECCR 116

Meeting times: MWF, 12 – 12:50

Office: Math 204

Office hours: calendar

## References

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.

- R. Hartshorne,
*Algebraic Geometry* - A. Grothendieck, Sur quelques points d'algèbre homologique
- R. Vakil,
*The rising sea* - J. Rotman, An introduction to homological algebra
- B. Fantechi, Stacks for everybody
- A. Altman and S. Kleiman,
*Introduction to Grothendieck duality theory* - J. Tate, Residues of differentials on curves
- S. Kleiman, The Picard scheme (in
*FGA Explained*) - A. Grothendieck,
*Technique de descente et théorèmes d'existence en géométrie algébrique*, Exposés V and VI - D. Mumford,
*Curves and their Jacobians*(published as an appendix to*The red book of varieties and schemes* - J.-P. Serre,
*Algebraic groups and class fields* - P. Deligne, La formule de dualité globale (Exposé XVIII in SGA4)
- M. Artin, Grothendieck topologies

## Syllabus

- The Picard group
- Stacks and Grothendieck topologies
- Read Stacks for everybody
- Read Olsson's lecture on stacks and/or Vistoli, §§3.1 and 4.1. You may need to refer back for definitions.
- Read SGA4, Exp. XVIII, §1.4, pp. 519–532 (original pagination)
- Read Artin's Grothendieck topologies, Chapter I, §0 and Chapter II, §§1 and 2.
- Lecture notes: Stacks
- Curves
- Reading: Hartshorne, §IV.1
- Reading: Altman & Kleiman, §VIII.1
- Exercises: Hartshorne, IV.1.1, IV.1.3, III.1.6, III.1.7, III.1.8
- Reading: Hartshorne, §§IV.2–5. As of Nov. 17, we have covered some of §§IV.2, IV.3, and IV.4 already. We will cover the group law in §IV.4 when we discuss the Jacobian after Thanksgiving.
- Exercises: Hartshorne §IV.2.1, IV.2.2 (how does this relate to our classification of genus 2 curves from class?),
**IV.2.5**, IV.2.7 (this is an important special case of the universal coefficients theorem),**classify all genus 3 curves**.**Submit the bold exercises on Monday, Nov. 27.** - Lecture notes: Curves
- October 6: coherent cohomology
- Reading: Hartshorne, §III.3 and §III.5
- Exercises: Hartshorne, III.3.1, III.3.2, III.4.1, III.4.7.
**Submit these on Monday, Oct. 16.** - Exercises: Hartshorne, III.5.1, III.5.2, III.5.3 (a), III.5.5
- Exercise: Hartshorne, III.3.7. Most of this will be done in class on Oct. 18. Finish the details and
**submit on Monday, Oct. 30**. You may want to make use of these exercises as lemmas. - Exercise: Prove Bézout's theorem.
**Submit on Monday, Oct. 30**(or later, if that's not enough time). - Lecture notes: Coherent cohomology
- September 29: Čech cohomology
- Reading: Hartshorne, §III.4
- Exercises: Hartshorne, §III.4, #4, 5, 6, 11. Exercises 4 and 11 will be discussed in class.
**Submit #5 and #6 on Monday, Oct. 9.**These exercises don't involve schemes. - Exercise: Hartshorne, §III.4, #10. This is really an exercise about torsors, not Čech cohomology. It will require some comfort with algebraic concepts from last semester, e.g., derivations and differentials.
- Lecture notes: Čech cohomology
- September 11: sheaf cohomology
- Reading: Hartshorne, §III.2
- Reading: Tohoku, Théorème 3.6.5 (this is Theorem III.2.7 in Hartshorne)
- Reading: for some background on homological algebra, see Rotman, §6.2.
- Exercise: Hartshorne, §III.2, #2.1, 2.2, 2.6.
- Exercise: Hartshorne, §III.2, #2.7. Do this using the Godement resolution and using torsors.
- Exercise: show that the category of sheaves of abelian group usually does not have enough projectives
- Lecture notes (updated): sheaf cohomology
- September 6: derived functors and sheaf cohomology
- Reading: Hartshorne, §III.1
- Reading: Tohoku, §1.10 (refer back for definitions, if necessary)
- Exercises: Hartshorne, §II.1, #1.16
- Exercise: Suppose that
*A*is a sheaf of rings on a topological space*X*. Show that the category of*A*-modules has enough injectives. - September 1: abelian categories, sheaves of abelian groups form one
- Reading: Tohoku, §§1.3, 1.4 or Vakil, §1.6 – 1.6.3
- Exercises: equvialence of definitions; Vakil, 1.6.B – D
- Lecture notes: abelian categories
- August 28: overview of the course, review of sheaves
- Reading: Hartshorne, §II.1
- Exercises: Hartshorne, §II.1, #1.2, 1.3 (a) (this asks you to prove that the definition of surjectivity given in class is equivalent to the one Hartshorne uses), 1.3 (b), 1.5, 1.8, 1.13, 1.15, 1.17, 1.18, 1.21, 1.22
- Lecture notes: sheaves

## Assessment

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.