Spring 2008
Professor: Dr. Hal Schenck
332 Illini Hall
Office hours: T-Th 120-220, and by appointment.
Phone: 333-2229.
E-mail: schenck@math.uiuc.edu.
Meeting times/rooms
T-Th 12-120pm, AH 141
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 be an introduction to algebraic geometry, which uses toric varieties as the main examples.
To be particularly down-to-earth, in modern algebraic geometry, a variety is constructed by gluing together affine varieties. In the toric case, an affine variety corresponds to a polyhedral cone, and gluing affine torics simply corresponds to gluing two cones together along a common facet. Another key concept in algebraic geometry is that of a divisor (a codimension one subvariety); in the toric case a divisor corresponds to an edge of the cone. In short, there is a beautiful dictionary between discrete geometric objects (cones, polytopes, etc.) and toric varieties.
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 (notes to be handed out), 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.
HW 1 Due 2/1
Chapter 1: 0.3, 0.9, 1.6,1.11,2.6,2.11,3.1,3.5