# Advanced Algebraic Geometry, Universiteit Leiden, Fall 2013

# News

- Below is a scan of the notes of lecture 1.
- Starting September 10 there is a problem session for this
course in room 401 from 14:45--15:30.
- To start reading the stacks project, follow the link to it in
the references below, and then click on ``browse the project online''.

## Teachers

Bas Edixhoven and David Holmes.
## Time and place

Tuesdays, lecture 15:45--17:30, problem session 14:45--15:30, all in
room 401.
### Course outline

The aim of this course is to state the Grothendieck Hirzebruch Riemann Roch
theorem for morphisms of smooth projective varieties over a field, and
prove as much of it as we can. This also means that the necessary
tools will be introduced:

- schemes (and their morphisms of course, and
properties thereof) (3 weeks),
- sheaves of modules, coherent, quasi-coherent, divisors and
Picard group (2 weeks)
- cohomology, higher direct images, some general homological
algebra, spectral sequences (2 weeks),
- Grothendieck groups of coherent sheaves (K
_{0}(X)) and
of locally free
coherent sheaves (K^{0}(X)) and their relation
(K(X):=K^{0}(X)=K_{0}(X)) and structures (1 week),
- intersection theory, Chow group, Chow ring A(X) (2 weeks),
- Chern ring morphism from K(X) to A(X)
_{Q}, and
the Todd class (1 week).

Once we have these tools available, the general Grothendieck
Hirzebruch Riemann
Roch theorem is easily stated. We follow the article by Borel and
Serre (see the list of references below).

In the remaining two weeks we will try to present as much as possible
of the proof given in Borel-Serre:

- first reductions,
- exactness and homotopy properties for K(X),
- proof for the projection from
**P**^{n} x X
to X,
- proof for Y ➝X the embedding of a codimension one
subvariety,
- blow up, and embeddings in higher codimension...

### References

- Borel-Serre
- Le théorème de Riemann Roch
- Hartshorne
- Algebraic geometry. Graduate Texts in Mathematics,
No. 52. Springer-Verlag, New York-Heidelberg, 1977.
- de Jong et al.
- Stacks project.

### Lecture notes

For David's notes, go to his home page.
## Exam

Here is the take home assigment. Note
that we want to receive the solutions by Sunday January 26, 23:59. We
will inform the students who registered for the oral exams on January
28 en 29 about the schedule.

Bas Edixhoven <edix@math.leidenuniv.nl>
Last modified: Tue Dec 10 17:52:33 CET 2013