Algebraic geometry plays a central role in mathematics. It is applied in many other branches such as algebraic topology, combinatorics, commutative and noncommutative algebra, number theory and representation theory.
Modern algebraic geometry is built on commutative algebra. Therefore the course would not be complete without covering the necessary parts in commutative algebra, but no previous knowledge in that field is assumed.
The course will start at the basics and then progress as far as posssible.
Prerequisite: Math 504-5-6 (or equivalent first-year graduate course in algebra).
Autumn Quarter: affine and projective varieties, regular and rational maps, Zariski topology, families, parameter spaces, correspondence between ideals and algebraic sets, Nullstellensatz, and many examples.
Winter Quarter: algebraic groups, dimension, Hilbert polynomials, tangent spaces, smoothness, degree, tangent cones, and many more examples.
Spring Quarter: to be determined - may include one or more of: classification of curves and surfaces, Groebner bases, advanced commutative algebra topics, or an introduction to sheaves and schemes.
Textbooks
| required: | Joe Harris, Algebraic Geometry, A First Course |
| recommended: | William Fulton, Algebraic Curves |