Mathematics 608G, Fall 2000

Course Description

The course will be an elementary course in algebraic geometry emphasizing practical methods for computer computation. The text will be the nominally ``undergraduate text'' by Cox, Little, and O'Shea. Since the book does aim at an undergraduate audience, there is much in the text that I will not need to cover. As a result, I plan to cover most of the book (all except the chapters on robotics and invariant theory). The outline of the course, from their chapter headings is

Elementary ideas of what a variety is (including parametrization).
Groebner Bases. I.e. how to compute in a polynomial ring over a field.
Elimination Theory. Including implicitization, singular points, envelopes, resultants and the extension theorem.
Algebra-Geometry Dictionary. Including the Hilbert Nullstellensatz, Zariski Closure, irreducible varieties, primary decomposition.
Functions on a Variety. Coordinate rings, rational functions, birational equivalence.
Projective Algebraic Geometry. Projective closure, elimination theory, example of quadric hypersurfaces.
Dimension of a Variety. Equivalent definitions, Hilbert Functions, singularity, tangent cone.

Amplifications of the material will be given when it seems appropriate. Methods for actually computing all of the above will be given. The two computer algebra packages I will emphasize will be Mathematica and CoCoA. I also suggest that the students look at Macaulay.

A new version of CoCoA (CoCoA 4.0) has just been announced. You may receive information on downloading this package and further information on CoCoA by consulting their homepage.

This course is not meant to give the students a complete introduction to algebraic geometry. For that the students should attend Prof. Ranachandran's Math 606. Taking both of these courses will give the student two very different perspectives on this important subject.

Prerequisites

Math 600-Math 601, one year of graduate level abstract algebra.

Class meeting time

Tuesdays and Thursdays from 9:30 to 10:45 in room B0425 in the Mathematics Building.

Instructor

The lecturer in MATH 608G is Professor William Adams. His office is in room 1111 of the Mathematics Building, and his university phone extension is 55056. (Calling from outside the university, call 301-405-5056.) He can also be reached by email at wwa@math.umd.edu.

Textbook

Ideals, Varieties, and Algorithms by D. Cox, J. Little, and D. O'Shea, published by Springer Verlag.

Bibliography

An Introduction to Grobner Bases by W. Adams and P. Loustaunau published by the American Mathematical Society (Graduate Studies in Mathematics, Vol 3).
An Introduction to Grobner Bases by R. Froberg published by Wiley-Interscience Series.
Using Algebraic Geometry by D. Cox, J. Little, and D. O'Shea, published by Springer Verlag.
Computational Commutative Algebra 1 by M. Kreuzer and L. Robbiano, published by Springer Verlag.
G.-M. Greuel, Computer Algebra and Algebraic Geometry- Achievements and Perspectives, Journal of Symbolic Computation, 30 (2000) 253-290.
D. Bayer and D. Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991), 1--48, Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993.