Mathematics Department - Math 535 - Algebraic Geometry I

Math 535 - Algebraic Geometry I

Specific Semesters

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 :

  1. Affine and projective space, hypersurfaces, rational and rationally connected varieties
  2. Morphisms, products, and projections
  3. Moduli spaces and families of varieties
  4. Grassmannian varieties and algebraic groups
  5. Dimension and Hilbert polynomials
  6. Smoothness and tangent spaces
  7. Degree of a variety
  8. 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.

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