|
Lecturer |
Prof. Dr. Paul Nelson |
Lectures |
Tue 15-17, HG E 1.2
Thu 10-12, HG G 26.5 |
Coordinator |
Claude Eicher |
Exercises |
Thu 15-16:30, LEE C 104
Fri 15:10-15:55, HG E 33.5 |
First lecture: Tuesday, February 17, 2015
First exercise class: Friday, February 20, 2015
We plan to cover some of the core topics concerning algebraic varieties over an algebraically closed field and then some additional ones, depending upon the pace to which the course settles.
The core topics include affine and projective varieties, morphisms between them, basic properties of such (irreducibility, dimension, finiteness, degree, ...), sheaves, schemes, and the functor of points perspective, some key geometric notions and constructions (birational maps, blowups, projections, divisors, ...), some key examples (Segre, Veronese, Grassmannian, ...), properties of images of morphisms (Chevalley's theorem, universal closedness of projective varieties, ...), differential notions (tangent space, smoothness, ...).
For the additional topics, I have in mind the motivating goal of developing the basic theory of curves and surfaces (particularly intersection theory on the latter) to the extent required to present in the final lectures Weil's proof of the Riemann hypothesis for curves over finite fields.
We'll loosely follow
skipping around a bit and referring as needed to other
sources such as
A course in commutative algebra is an official prerequisite, and it will certainly be helpful if you have (for instance) read
and done many of the exercises, but should not be necessary for general comprehension if you are willing to accept certain facts as black boxes.
We record below some reading assignments intended to complement the lectures.
Week 1 |
p. 1-4 and 14-20 in Hartshorne, p. 17-22 in Harris (skip discussion of projective varieties for now) |
Week 2 |
p. 4-7 and Lemma 4.2 in Hartshorne, p. 48-49 and 61-62 in Harris |
Week 3 |
p. 8-11 (projective varieties) and Lemma 4.1 in Hartshorne, section 5 (varieties) in Gathmann |
Week 4 |
p. 3-15 in Harris |
Week 5 |
section 4 of Chapter I in Hartshorne |
Week 6 |
p. 72-79 in Harris |
Week 7 |
section 9 (birational maps and blowing up) in Gathmann |
Week 10 |
Gathmann p.62 up to Remark 7.26, Milne sections 7a-7d |
Week 11 |
section 6 of Chapter I in Hartshorne |
Week 13 |
sections II.4-II.5 in The Arithmetic of Elliptic Curves, J. H. Silverman |
Week 14 |
sections II.1-II.2 in Hartshorne |
can be found here. (Updated 21 Aug)
Problem sheet | Due on date |
Sheet 1 |
27.2.2015 |
Sheet 2 |
6.3.2015 |
Sheet 3 |
13.3.2015 |
Sheet 4 |
20.3.2015 |
Sheet 5 |
2.4.2015 |
Sheet 6 |
17.4.2015 |
Sheet 7 |
30.4.2015 |
Sheet 8 |
8.5.2015 |
Sheet 9 |
15.5.2015 |
Sheet 10 |
22.5.2015 |
Sheet 11 |
29.5.2015 |
Selected Solutions to * Problems (A. Puttick) |
Group | Room | Assistant | Time |
1 |
LEE C 104 |
Alexandre Puttick |
Thursday, 15-16:30 |
2 |
HG E 33.5 |
Claude Eicher |
Friday, 15:10-15:55 |
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