Math 567, Spring 2008
Topics in algebraic geometry:
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.,
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.
- How many lines meet four general lines in 3-space?
- How many conics are tangent to five plane conics?
- 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,
- Riemann-Roch formula
- Equivariant intersection theory
- Localization and the Bott residue formulae
Prerequisites: Familiarity with basic algebraic geometry
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.
Herman Brown 402
Office hours: TBA