Mathematics 535 -- Introduction to Algebraic Geometry -- Fall 2002
This course will be an introduction to the study of algebraic
varieties, that is the zero sets of polynomials in several variables.
Just as linear algebra has geometric content of lines, planes
and hyperplanes as well as the algebraic structure of vector spaces,
subspaces and linear maps the subject of algebraic geometry has
simultaneously the geometric flavor of surfaces, hypersurfaces, etc. and
the algebraic structure of commutative algebra of rings of polynomial
functions. The subject is the study of the interplay of these two points
of view.
The emphasis of the course will be on examples of algebraic varieties and
general attributes of varieties and morphisms as reflected in these
examples. Examples of algebraic varieties arise in many places in
physics, topology, geometry, combinatorics and number theory and the examples studied
in this course will be often be drawn from other areas of mathematics.
I plan to concentrate on the geometrical aspects of the subject, which
is where the classical beginnings lie, and to bring in the algebraic
aspects as we accumulate examples.
Topics will be drawn from the following :
- Affine and projective space, hypersurfaces, rational curves
- Morphisms, products, and projections
- Moduli spaces and families of varieties
- Grassmannian varieties and algebraic groups
- Dimension and Hilbert polynomials
- Smoothness and tangent spaces
- Degree of a variety
- Algebraic curves
Prerequisites: Basics of linear algebra, rings, and fields.
The standard graduate algebra course is sufficient.
Text: Algebraic Geometry, a First Course, by J. Harris, Springer
Graduate Texts in Mathematics 133 ISBN 0-387-97716-3, 1995.
This text and additional references will be placed
on reserve.
Course Format: There will be weekly homework assignments. Each student
will adopt a family of algebraic varieties and report on their
basic properties and special quirks.
More Information: Contact J. Tunnell in Hill 546, email to
tunnell@math or examine
the course web site.