
Lecturer 
Prof. Dr. Paul Nelson 
Lectures 
Tue 1517, HG E 1.2
Thu 1012, HG G 26.5 
Coordinator 
Claude Eicher 
Exercises 
Thu 1516:30, LEE C 104
Fri 15:1015:55, 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. 14 and 1420 in Hartshorne, p. 1722 in Harris (skip discussion of projective varieties for now) 
Week 2 
p. 47 and Lemma 4.2 in Hartshorne, p. 4849 and 6162 in Harris 
Week 3 
p. 811 (projective varieties) and Lemma 4.1 in Hartshorne, section 5 (varieties) in Gathmann 
Week 4 
p. 315 in Harris 
Week 5 
section 4 of Chapter I in Hartshorne 
Week 6 
p. 7279 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 7a7d 
Week 11 
section 6 of Chapter I in Hartshorne 
Week 13 
sections II.4II.5 in The Arithmetic of Elliptic Curves, J. H. Silverman 
Week 14 
sections II.1II.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, 1516:30 
2 
HG E 33.5 
Claude Eicher 
Friday, 15:1015:55 
Wichtiger Hinweis:
Diese Website wird in älteren Versionen von Netscape ohne
graphische Elemente dargestellt. Die Funktionalität der
Website ist aber trotzdem gewährleistet. Wenn Sie diese
Website regelmässig benutzen, empfehlen wir Ihnen, auf
Ihrem Computer einen aktuellen Browser zu installieren. Weitere
Informationen finden Sie auf
folgender
Seite.
Important Note:
The content in this site is accessible to any browser or
Internet device, however, some graphics will display correctly
only in the newer versions of Netscape. To get the most out of
our site we suggest you upgrade to a newer browser.
More
information