MATH 7350: Topics in Algebra: Intersection theory (Fall 2010)

Instructor: Michael Stillman

Prerequisites: some knowledge of algebraic geometry, preferably some schemes, sheaves, cohomology.

Textbook: We will use the forthcoming book by Eisenbud and Harris on intersection theory (the name of the book is not yet finalized).

This is an intermediate course in algebraic geometry, centered around enumerative geometry and intersection theory of mostly non-singular varieties. Many examples, and applications of the general apparatus of algebraic geometry, will be given. In some ways, the central value of this course is seeing algebraic geometry in action. On the other hand, enumerative geometry is a beautiful field, full of many interesting examples and techniques. (Many of these methods are implemented in Macaulay2 as well, so more in depth examples are possible, and we will include such examples).

Potential topics include (these are roughly the chapters of Eisenbud-Harris. Each chapter has a wealth of examples, applications, and exercises):

  1. Chow groups — rational equivalence, proper pushforward
  2. Intersection products
  3. Intersection theory on Grassmannians, Schubert classes
  4. Vector bundles and Chern classes — linearizing non-linear behavior
  5. Example: Lines on hypersurfaces (e.g., 27 lines on a cubic surface)
  6. Singular elements of a linear system (applications include Pluecker formulas)
  7. Parameter spaces (e.g., complete conics)
  8. Chow rings and projective bundles, Segre classes
  9. Tangent bundles of projective bundles (application: contact bundles)
  10. Degeneracy loci of maps of vector bundles, Porteous' formula
  11. Excess intersection
  12. Groethendieck-Riemann-Roch theorem and applications
  13. Brill-Noether theory of curves
  14. Fulton's theory: deformation to the normal cone.

The book (and also this course) is organized such that there are interesting explicit enumerative geometry problems that will be solved in each chapter.

The amount that we cover of these topics will depend on the background and interest of those taking the course.