Math 633 Algebraic Geometry
Fall 2002

Instructor: Ali Sinan Sertöz, Department of Mathematics
Room: SA-121, Phone: 1490
Office Hours: Tuesday 13:40-15:30

Text: Robin Hartshorne, Algebraic Geometry, Springer-Verlag 1977.

TUESDAY 10:40-11:30 SAZ-01
THURSDAY 10:40-12:30 SAZ-01



1 24 Sep-26 Sep Affine and projective varieties, morphisms:   I.1, I.2, I.3
2 1 Oct-3 Oct Rational maps, Blowing up, nonsingular varieties: I.4, I.5
3 8 Oct-10 Oct Intersection in projective space: I.7
4 15 Oct-17 Oct Sheaves and schemes: II.1, II.2
5 22 Oct-24 Oct Main properties of schemes: II.3, II.4, II.5
6 31 Oct Divisors and projective morphisms: II.6, II.7
7 5 Nov-7 Nov Differentials: II.8
8 12 Nov-14 Nov Cohomology: III.1, III.2, III.3
9 19 Nov-21 Nov Cech Cohomology and examples: III.4, III.5
10 26 Nov-28 Nov Serre duality, flat, smooth and etale morphisms: III.6, III.7, III.8, III.9, III.10
11 3 Dec Zariski's main theorem: III.11
12 10 Dec-12 Dec Riemann-Roch and Hurwitz's theorems: IV.1, IV.2
13 17 Dec-19 Dec Embeddings in projective space, Elliptic curves: IV.3, IV.4
14 24 Dec-26 Dec The canonical embedding: IV.5
15 31 Dec-2 Jan Classification of space curves of low degree: IV.6

Note on the syllabus:
The syllabus is over ambitious if taken literally. However we will cover only the main theorems and definitions which are necessary to understand the chapter on curves. Some proofs will be done in detail but most theorems will be taken on faith and demonstrated on examples. Hartshorne's book has the status of being the book of introduction to algebraic geometry. However it is an inside information that one can spend a life time trying to master Hartshorne's book! This is clearly not desirable. We have two aims in mind for this course: (1) to acquaint ourselves with the basic ideas and terminology so that we can refer back to Hartshorne's book any time later to pick up a definition or a theorem without much back reading, and (2) to study the ideas on curves in as much detail as time permits. If there is public demand and pressure I may open up a reading course on surfaces, covering chapter V or equivalent, next semester.

How to study:
Read the text lightly first. Then study the Examples. Construct your own examples and write them out explicitly. Test all new theorems against your examples to check if  they make sense. Check all conjectures, guesses and new ideas against your examples. Read all the exercises at the end of the section. Attack some of them. Then study some of the important proofs. Discuss with your friends to find out what they make of geometry!

There are 21 homework problems, each is 10 points. There will be 2 midterm   and  1 final takehome exams, each is 30 points.  Midterm and final exam problems will be chosen from the harder or longer exercises of the book. In all homework and takehome work you are expected to talk, discuss and argue if necessary with your classmates for the correct solution. You are also free to consult books. But you must write your solution on your own.
Homework Problems: (70%)
Chapter I:   Exercises 1.8, 2.14, 5.1, 5.2, 5.3, 7.2.   Due on 15 October 2002 Tuesday.  Solutions
Chapter II:  Exercises 1.18, 1.19, 1.21, 6.5, 8.4. Due on 12 November 2002 Tuesday.    Solutions of 1.18, 1.19, 1.21, 6.5
Chapter III: Exercises 4.3, 5.3, 5.5, 7.3, 10.3. Due on 10 December 2002 Tuesday.
Chapter IV: Exercises 1.7, 2.2, 3.6, 5.1, 6.5. Due on 3 January 2003 Friday.
Midterm I: (10%) Due on 31 October 2002 Thursday.
Midterm II: (10%) Due on 28 November 2002 Thursday.
Final: (10%) Due on 13 January 2003 Monday.