Algebraic Geometry, Spring 2005

For a general description of this course, see the description in the `studiegids', and for the schedule (time and location) see schedules in the `studiegids', or see the pages of the Dutch Master Program in Mathematics.

This page gives a progress report on what has been done (course and problem session), and gives a planning of what is to follow. And there is more: the New Deal regarding the procedure for getting a grade for this course. This procedure, of which the organisational details still have to be worked out, is that those who wish to obtain a grade pass an oral examination, during which the selected exercises for the problem session will be discussed, with all documentation (the book, other books, solutions) available.

1. February 7
In the course, section I.1 (of Hartshorne's book) has been treated up to Remark 1.4.6. A proof of Hilbert's Nullstellensatz has been promised for later (and maybe a proof of Hilbert's Basis Theorem, saying that if A is noetherian, then A[x] is noetherian). The selected exercises are: 1.1(a),(b), 1.2, 1.3, 1.4, 1.6, 1.7.

2. February 14
The rest of section I.1, up to Corollary 1.6, has been discussed (for the moment, we do not care about dimension). Section I.2 has been discussed, again omitting everything about dimensions. The Zariski topology on PN has been defined as a quotient topology rather than as zero sets of homogeneous polynomials; here are the notes typed by Peter Bruin for this other treatment (comments are welcome!). Selected exercises: 2.9, 2.10(a) and (b), 2.11, 2.14 and 2.15. Hint for 2.14: give left-inverses for Ψ on the standard opens of PN.

3. February 21
Section I.3 has been discussed up to the definition of morphisms of varieties, and Lemma 3.6 has been treated. As an application, the Cayley-Hamilton theorem was proved using the principle of ``regular continuation''. Selected exercises: 3.2, 3.3, 2.12, 3.4, 3.5.

4. February 28
Section I.3 has been treated up to Thm. 3.4. The necessary results on localisation in an integral domain have been explained. Selected exercises: 3.6, 3.7, 3.1, 3.19.

5. March 7
Proposition I.3.5 has been treated. A counter example to exercise I.3.19**(b) has been given for all n at least 1; one simply uses that for p prime, the derivative of x+xp is 1, if the base field is of characteristic p. Chapter II, Section 1 has been treated up to Proposition 1.1, and a proposition on sheaves restricted to a basis of the topology has been given; see the notes typed by Peter Bruin, or see Grothendieck, page 25 of the IHES edition of EGA I (the file in djvu format is much smaller than the pdf file). Selected exercises (Chapter II): 1.1, 1.9, 1.22, 1.14 and: show that the maps q:An+1-{0}---->Pn are quotient maps for the kx-action in the category of varieties.

6. March 14
The remainder of Section II.1 has been treated, except the last two definitions. The notions of products and coproducts in categories, additive categories, kernels, cokernels and abelian categories have been defined. The results in the book have been summarized as follows: the categories of sheaves and presheaves on a topological space are abelian, taking stalks commutes with taking kernels and cokernels, and a sequence of sheaves is exact iff it is exact on all stalks. The example of the topological space {-1,0,1} with -1 and 1 closed and 0 generic (its closure is everything) has been given in detail. Selected exercises: 1.3, 1.14, 1.16, 1.17.

7. March 21
The last two definitions of II.1 have been treated, and an example has been given. Section II.2 has been treated up to the statement of Proposition 2.2. Here are the exercises; they have not been taken from the book, but have been typed by Theo, and have been handed out.

March 28
No lecture: easter time.

8. April 4
Proposition 2.2 will be proved, ringed spaces and schemes will be defined, and examples will be given. Selected exercises: 2.1, 2.2, 2.4, 2.8, 2.12.

9. April 11
The Proj of a graded algebra has been constructed, and section II.3 has been treated up to the proof of Prop. 3.2. Here is a construction of Proj in ps format, a bit different from the one in the book. Here are the exercises; they have been taken from the book, essentially, but give some hints. They have been typed by Theo.

10. April 18
The remainder of Section II.3 has been treated. Exercises: 3.1, 3.5, 3.10 and some more typed by Gabor.

11. April 25
The notions of integral dependence and normalisation will be treated. Hilbert's Nullstellensatz will be proved, as well as certain results on the notion of dimension of schemes. This piece does not correspond to a part of the book. Here are the exercises, by Gabor. Here are the notes, typed by Peter Bruin.

12. May 2
The lecture was given by Gabor and Theo. Notions: separated and proper morphisms of schemes. Here are the exercises, by Theo and Gabor.

13. May 9
There will be no problem session, but only lecture, from 14:00 until 16:45. Subject: the rest of dimension theory for finitely generated algebras over a field (not in Hartshorne's book), sheaves of modules, coherent and quasi-coherent (section II.5 of Hartshorne's book), and divisors and Picard groups (section II.6).

May 16
No lecture: 2nd day of pentecost.

14. May 23
Last lecture. Subject: Picard groups, divisors, intersection theory on regular proper surfaces over a field. Here are the exercises, by Theo and Gabor. And here is the evaluation form for this course.

15. May 30
Only problem session (14:00-16:45), no lecture, no new problems, but an ideal opportunity to ask questions.

Schedule oral examinations

The oral examinations will take place in Leiden, at the Department of Mathematics, in Edixhoven's office (room 236), in the week of June 20. During the examination, the homework will be discussed, with all desired documentation available. Those who for some reason wish to postpone the examination can make an appointment (by email) with Edixhoven for August 2, 3, 4 or 5, or for another date if there are good reasons for that. During the last two sessions of the course/problem session it will be possible to choose an entry in the table below.

June 20June 21 June 22June 23June 24
9:00-9:30 ? XXX ? ? Janne Kool
9:45-10:15?XXX?? XXX
10:30-11:00 ? XXX Marco Streng ? XXX
11:15-11:45 Peter Bruin XXX XXX Kirsten Valkenburg XXX
12:00-12:30 XXX XXX JMC de Boer ? XXX
lunch
13:30-14:00 Jeroen Sijsling XXX Maarten Hoeve ? XXX
14:15-14:45 Rozenn Prodhomme Frank de Zeeuw Leo Kool ? XXX
15:00-15:30 ? Remkes Kooistra ? ? ?
15:45-16:15 ???XXX?
16:30-17:00 ???XXX?

Exceptions:




Bas Edixhoven <edix@math.leidenuniv.nl>
Last modified: Thu Aug 18 09:34:24 CEST 2005