Intersection Theory

Intersection theory is one of the oldest aspects of classical algebraic geometry, forming an essential component of enumerative geometry, the science of counting solutions to algebro-geometric problems such as the number of bi-tangents to a plane curve of degree d. In spite of its ancient roots, intersection theory was not put on a firm foundation until late in the 20th century. A number of different approaches succeeded, but perhaps the most elegant and far-reaching was the one developed by Fulton and described in detail in his book Intersection Theory. Besides giving a solid foundation for enumerative geometry, Intersection Theory introduced a number of new ideas and approaches which were subsequently taken over for use in many other areas of algebraic geometry.

In this course, we will cover the basics of Fulton's intersection theory and give a taste of some of its many applications. 

The course is designed for Ph.D. students and sufficiently advanced Masters' students,  although all interested parties are welcome to attend. The background required is the algebraic geometry and commutative algebra contained in for example Hartshorne's Algebraic Geometry chapters I and II (more or less), in particular, we will not need to use anything about cohomology of coherent sheaves, although we will need some facts from ``local" homological algebra, such as the functors Tor_i. If you are interested in the course but are unsure if you have the necessary prerequites, please see me to discuss this. Masters students should speak with me to discuss how the course will be graded.

We will meet twice weekly:

Monday, 12-14 (c.t.) T03 R03 D89

Wednesday, 10-12 (c.t.) T03 R03 D89

The first meeting will be on Wed. April 11

Text: Fulton, William, Intersection theory Second edition. Springer-Verlag, Berlin,  1998