> Algebraic Geometry - SS 12

AG Algebra und Geometrie

# Content:

## Overview

Most of mathematics is concerned at some level with setting up and solving various types of equations. Algebraic geometry is the mathematical discipline which handles solution sets of systems of polynomial equations. These are called algebraic sets.

Making use of a correspondence which relates algebraic sets to ideals in polynomial rings, problems concerning the geometry of algebraic sets can be translated into algebra. As a consequence, algebraic geometers have developed a multitude of often highly abstract techniques for the qualitative and quantitative study of algebraic sets, without, in fact, considering the equations at the first place. Modern computer algebra algorithms, on the other hand, allow us to manipulate the equations and, thus, to study explicit examples. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, now also adds to the toolbox of the algebraic geometer.

In this lecture, we discuss some of the basic operations in geometry and describe their counterparts in algebra. We also indicate how the operations can be carried through using computer algebra methods, and give a number of explicit examples, worked out with the computer algebra system SINGULAR. In this way, the lecture connects nicely to courses on commutative algebra and computer algebra.

## Dates

• Lectures:  (KIS entry)
• Mo, 10:00 - 11:30, 48-438, start: 16.04.2012
• Th, 10:00 - 11:30, 48-438, start: 19.04.2012
• Exercise classes  (KIS entry)
• We, 13:45 -15:15, 48-438, start: 18.04.2012

## Homework

Nr.

Notes

1

sheet 1

2

sheet 2

3

sheet 3

Typo in the definition of the Galois group in Exercise 11 fixed: May 06, 2012

4

sheet 4

5

sheet 5

6

sheet 6

7

sheet 7

8

sheet 8

9

sheet 9

10

sheet 10

11

sheet 11

12

sheet 12

13

## Credit points

Regular, active and successful participation in the exercise classes is required to achieve an "Übungsschein". This includes scoring at least 40% of the exercise points and presenting own solutions on the black board.
You have to pass an oral exam at the end of the term in order to get credit points for this lecture.
If you are in doubt whether or not you need an Uebungsschein, please ask your academic advisor ("Fachstudienberater" in German). In case you study mathematics, you may find him here.

## Literature

1. David A. Cox, John B. Little, and Donal O'Shea: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer 2007
2. Miles Reid: Undergraduate Algebraic Geometry. London mathematical society, student texts, vol.12
3. Igor R. Shafarevich: Basic Algebraic Geometry 1: Varieties in Projective Space. Springer 1994
4. Robin Hartshorne: Algebraic Geometry. Springer 1997
5. Wolfram Decker, Frank-Olaf Schreyer: Varieties, Gröbner bases, and algebraic curves, book in preparation.
6. Wolfram Decker, Gerhard Pfister: A First Course in Computational Algebraic Geometry, to appear.

## Lecture notes

Last change: July 23

Wolfram Decker

 July 23, 2012 [top]