### Inhalt des Dokuments

### zur Navigation

# Algebraic Geometry 2

This advanced course is a continuation of the course in Algebraic Geometry from the summer term. The contents of said lecture are therefore prerequisite for attending. It will also be announced as a BMS advanced course.

The course will start on the 22nd of october.

**Important information: **There will be an Algebraic Geometry 3 lecture during the upcoming summer term. It will be possible to have** one exam for** **both lectures**.

**Important information 2: **There will be an **additional lecture **on wednesday **december 9th**. There will be **no lecture** on thursday **december 17th**.

**Important information 3: **You can now apply for an **oral exam. **Possible dates are **february 17th (2pm-6pm) and 18th (8am-2pm)**. Please come to MA 319 or write an e-mail to breiding-at-math.tu-berlin.de, if you want an exam.

**Important information 4: **The **next date for exams** is **april 19th (2pm-6pm)**. Please come to MA 319 or write an e-mail to breiding-at-math.tu-berlin.de, if you want an exam.

### Attention

Students planning to write a Master's Thesis under supervision of Prof. Peter Bürgisser are expected to attend this lecture.

# Lecture times

Thursday | 10:15-11:45 | MA 549 | Prof. Dr. Peter Bürgisser |

There will be a lecture on **Wednesday**, **December 9th**, from **8:15 - 9:45** in **MA-316**.

## Contents

The prospective contents of this lecture will be the following:

(see also Shafarevich, *Basic Algebraic Geometry*, Volume I)

**Chapter 1. Local Properties**

- Formal power series
- Repetition: local rings and tangent spaces
- Local parameters
- Power series expansion
- Local equations

**Chapter 2. Birational maps**

- Blowup of projective spaces
- Local blowup
- Behaviour of subvarieties under blowup

**Chapter 3.****Normal varieties**

- Normal varieties
- Normalization of varieties
- Projective embedding of smooth varieties

**Chapter 4. Fibers of regular maps**

- Nonsingularity of fibers
- Ramification
- Irreducibility