Text: Algebraic Geometry, a First Course, by J. Harris, Springer, Graduate Texts in Mathematics 133 ISBN 978-0387977164
Basics of linear algebra, rings, and fields. The standard graduate algebra course is sufficient. Basics of elementary topology will be assumed.
This course will be an introduction to the study of algebraic varieties, that is the zero sets of polynomials in several variables. In linear algebra the geometric content of lines, planes and hyperplanes interacts with the algebraic structure of vector spaces, subspaces and linear maps. Similarly 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, that the rings of functions on a space are paramount in understanding its geometry, and that any commutative ring can be considered as the ring of functions on an algebraic geometric object. The emphasis of the course will be on examples of algebraic varieties and schemes and general attributes of varieties,schemes 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, schemes
- Categories, Morphisms, Products, and Projections
- Families of varieties and the Grassmannian variety
- Dimension and Hilbert polynomials
- Smoothness and tangent spaces
- Degree of a variety