MA 764 - Algebraic Geometry (Fall 2002)

Course by Uwe Nagel at the University of Kentucky.

Overview

Schedule

MWF, 10:00 - 10:50, CB 349

Material

Algebraic Geometry is the study of sets of common zeros of polynomials, which are called algebraic varieties. We will begin by introducing basic varieties and constructions such as affine and projective varieties, regular and biregular maps. Then we will discuss dimension, degree, genus, tangent space, smoothness, and families of algebraic varieties. Finally, we hope to give some ideas how to get an overview over the set of algebraic varieties.

The course will be as self-contained as possible though some familiarity with the ring-theoretic concepts of a first year graduate Algebra course is preferable. The basic reference for the course will be Harris' book.

The grade will be based on active participation in the course and the quality of the homework.

CONTENTS:

2. Ideals and varieties

3. Regular functions and maps

4. Ideals and schemes

5. Examples of determinantal schemes

6. Hilbert function and Hilbert polynomial

7. Bezout's theorem

8. Tangent spaces and smoothness

9. Grassmanians

10. Hilbert schemes

11. Linkage of schemes and ideals

12. Even liaison classes of space curves

Homework

Set 1 (pdf-file, ps-file)
Set 2 (pdf-file, ps-file)
Set 3 (pdf-file, ps-file)
Set 4 (pdf-file, ps-file)
Set 5 (pdf-file, ps-file)

Literature

D. Eisenbud, Commutative Algebra with a view towards Algebraic Geometry.

D. Eisenbud, J. Harris, Schemes: The Language of Algebraic Geometry.

J. Harris, Algebraic Geometry.

J. Migliore, Introduction of Liaison Theory and Deficiency Modules.

Last Up-date: December 10, 2002

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