MA 764 - Algebraic Geometry (Fall 2004)
Course by Uwe Nagel
at the University of Kentucky.
Overview
Schedule
MWF, 11:00 - 11:50, CB 347
Material
Algebraic geometry is the the study of sets of common
zeros of polynomials which are called algebraic varieties. We will
begin by introducing fundamental concepts and constructions such as
affine and projective varieties, regular and birational maps. Then
we will discuss dimension, degree, tangent spaces, and
smoothness. Finally, we hope to give
some ideas on how to get an overview over the set of algebraic
varieties. Curves and toric varieties will serve as motivating and
illustrating examples.
The course will be as self-contained as possible though some
familiarity with the ring-theoretic concepts of a first year
graduate Algebra course (MA 561/661) is preferable. The basic reference for the
course will be Hulek's book.
The grade will be based on active participation in the
course and the quality of the homework.
CONTENTS:
1. Affine varieties
2. The Zariski topology
3. Polynomial functions and maps
4. Rational maps and localization
5. Projective varieties
6. Projective varieties and ideals
7. Rational maps and morphisms
8. Ideals and schemes
9. Hilbert functions and Hilbert polynomials
10. Bezout's thoerem
- D. Eisenbud, Commutative Algebra with a
view towards Algebraic Geometry.
- D. Eisenbud, J. Harris, Schemes: The Language
of Algebraic Geometry.
- J. Harris, Algebraic Geometry.
- K. Hulek, Elementary algebraic geometry
- J. Migliore, Introduction of Liaison
Theory and Deficiency Modules.
- K. E. Smith, L. Kahanpää, P. Kekäläinen,
W. Traves, An invitation to algebraic geometry.
Eric Stokes: Every variety in n-space is the intersection of
n hypersurfaces.
Rachelle Bouchat: What makes a complex exact? (word file)
Dibyajyoti Deb: The syzygy problem.
Sonja Petrovic: Gröbner bases and computing syzygies. (pdf file)
David Watson: Elliptic curves and Weierstrass p-function. (word file)
Tricia Muldoon: Geometric consequences of maximal growth of the
Hilbert function.
Julia Chifman: Phylogenetic algebraic geometry.
Daniel Pinzon: Why schemes?
Last Up-date: Nov 4, 2004