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.

**Schedule:**

TUESDAY | 10:40-11:30 | SAZ-01 |

THURSDAY | 10:40-12:30 | SAZ-01 |

**Topics: **

- Affine and projective varieties. (Chapter I)
- Sheaves and schemes. (Chapter II)
- Cohomology. (Chapter III)
- Curves. (Chapter IV)

**Syllabus:**

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

**Grading:
**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.

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.