Math 567, Spring 2008
Topics in algebraic geometry: Intersection Theory

MWF 9:00AM, HB 453

This is an introduction to the techniques of modern intersection theory in algebraic geometry. There will be a strong emphasis on applications to enumerative geometry and parameter spaces, e.g.,

How many lines meet four general lines in 3-space?
How many conics are tangent to five plane conics?
The first half of the semester will be a rapid survey of general techniques, based loosely on the first nine chapters of the Fulton's book. While the basic constructions and definitions will be explained carefully, we will not prove every functoriality statement needed to make the theory run. The second half of the semester will be devoted to applications and extensions, especially those developed over the last decade or so.
Cycles, rational equivalence, proper push-forward, flat pull-back, basic exact sequences
Cartier and Weil divisors, intersecting with divisors
Vector bundles, projective bundles, and Chern classes
Segre classes and normal cones
Intersection products, Gysin homomorphisms, blow-ups
Intersection multiplicities
Nonsingular varieties, intersection ring, Bezout Theorem
Excess intersection, double point formulas
Grassmannians, degeneracy loci, determinantal formulas, Thom-Porteous formulas, Schubert calculus
Riemann-Roch formula
Equivariant intersection theory
Localization and the Bott residue formulae

Prerequisites: Familiarity with basic algebraic geometry

Reference book: W. Fulton, Intersection theory, Springer Verlag 1998

Weekly problem sets (70%): Due each Wednesday starting January 16. They should be turned in during class. Homework assignments are not pledged. You are strongly encouraged to work together, though each student should write up her/his own submission.

Presentation (30%): A public presentation of a research paper using techniques developed in class.

Any student with a documented disability seeking academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All such discussions will remain as confidential as possible. Students with disabilities will need to also contact Disability Support Services in the Ley Student Center.

Brendan Hassett
Herman Brown 402
(713) 348-5261
Office hours: TBA