Fall 2009
Professor: Dr. Hal Schenck
332 Illini Hall
Office hours: T-Th 150-250, and by appointment.
Phone: 333-2229.
E-mail: schenck@math.uiuc.edu.
Meeting times/rooms
T-Th 1230-150pm, Everitt 170
Course Description. Toric varieties are objects at the interface of algebra, geometry and combinatorics. They can be studied from any one of these viewpoints, but it is the interplay between viewpoints that makes them so interesting. This course will cover the following broad topics, from a toric perspective: Quotient Constructions, resolution of singularities, Chow ring, Hirzebruch-Riemann-Roch and Minimal model program. In order to make the course available to all, I will begin the semester with a turbo two week review of the key constructions from the spring 2009 class on toric varieties.
A more detailed description of topics:
1. Homogeneous coordinates: Quotient constructions in algebraic
geometry, Quotient construction in toric case, Cox ring, connections
to polytopes.
2. Toric surfaces: resolution of singularities, continued fractions,
Grobner fan and Mckay correspondence, first glimpse of Riemann-Roch.
3. Cohomology: Chow ring, Hirzebruch-Riemann-Roch and possibly
relative version (Grot-Hirz-Riemann-Roch).
4. Topology of toric varieties, moment maps.
5. Toric Minimal Model Program.
The prerequisite of the class is a class in commutative algebra, at the level of Atiyah-Macdonald, although motivated students who know undergraduate algebraic geometry at the level of Cox, Little, O'Shea ``Ideals,varieties, and algorithms'' may be able keep up. The main objective of the class is to bring the abstract concepts of modern algebraic geometry to life with lots of examples.
Text. Toric Varieties, Cox, Little, Schenck.
Grading. Your grade will be determined by class participation and homework, which will be collected every two weeks. I expect that the amount of time you spend outside of class on this course to be about 5 hours/week, with 3 hours devoted to reading before class, and 2 hours to working on suggested problems.