Mathematics 535 -- Introduction to Algebraic Geometry -- Fall 2008
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. We will explore several themes of modern
algebraic geometry - that algebraic varieties arise in many areas of
mathematics, that families of algebraic varieties are often algebraic
varieties themselves, and that the functions on a space are paramount
in understanding its geometry.
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 and
rationally connected varieties
- 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
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.