Here's the official description of the course, from the undergraduate catalog. It is quite consistent with my own ideas about the course.
Math 5385 Introduction to Computational Algebraic Geometry
(4 cr; QP_3251 or equiv; SP_2263 or equiv)
Geometry of curves and surfaces defined by polynomial equations.
Emphasis on concrete computations with polynomials using computer
packages and on the interplay between algebra and geometry. Abstract
algebra presented as needed; no algebra prerequisite.
The course will be based on the book Ideals, Varieties and Algorithms,by Cox, Little, and O'Shea. In one semester we can cover the first four chapters, along with part of either chapter 5 or chapter 8. The course should be accessible to students with a reasonably solid background from sophomore level mathematics - either the standard or IT version.
We'll start by introducing some basic stuff about affine varieties - particularly curves
in 
2 and surfaces and curves in
3. We'll mention briefly the parallel concepts in 
2 and
3. This leads to a discussion of
ideals in the corresponding polynomial rings. (It's fairly concrete at this stage; we don’t have to talk about the abstract concept of ring until later.) We will use suitable graphics software in the computer labs to help with the visualization of implicitly defined curves and surfaces.
The method of Groebner bases provides an effective method to determine whether a polynomial is an element of a given ideal. This can be viewed as the multivariable analogue of long division of polynomials in one variable. Computations can be done by hand at first, but we will learn how to use a computer algebra package - probably Maple - to study more interesting examples. We will then apply Groebner basis methods to study other topics, such as projections of varieties, finding the implicit equations of a variety given in parametric form, and finding the singular points of a variety.
After developing some proficiency with computational methods, we will apply them to develop some introductory parts of the theory of algebraic geometry. At the end of the semester, we'll have a short introduction to either projective algebraic geometry [chapter 8] or polynomial and rational functions on a variety [chapter 5].
In summary: at the beginning we will assume knowledge only of sophomore level linear algebra and polynomials in several variables, including partial derivatives. Treating the subject in a fairly concrete way and moving forward in a systematic way from this starting point, we will reach a level where students can understand some of the theory in algebraic geometry. Thus, the course will not be organized along the lines of a standard graduate level algebraic geometry course.
For math majors: Math 5385 is on the Algebra List for the
Upper Division
Math Requirements
For further information: Please send me an e-mail or call me at the phone number listed below.
Back to the class homepage.
Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail:
roberts@math.umn.edu
http://www.math.umn.edu/~roberts