Department of Mathematics

Algebraic Geometry

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Lecturer Prof. Dr. Paul Nelson
Lectures Tue 15-17, HG E 1.2

Thu 10-12, HG G 26.5

Coordinator Claude Eicher
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.

Reading Assignment

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

Lecture Notes

can be found here. (Updated 21 Aug)


Problem sheet Due on date
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Sheet 6
Sheet 7
Sheet 8
Sheet 9
Sheet 10
Sheet 11
Selected Solutions to * Problems
(A. Puttick)

Exercise groups

Group Room Assistant Time
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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