Meeting days/times and credit: Lectures Monday and Wednesday, 930-1045, class will run July 14-August 15. Room: Parker Hall 320. Office hour after class Monday and Wednesday 11-12. Credit variable, grade determined by HW, make a decent effort and you'll get an A.
Course Description:
Toric varieties are algebraic varieties defined by combinatorial data, and there is a wonderful interplay between algebra, combinatorics and geometry involved in their study. Many of the key concepts of abstract algebraic geometry (for example, constructing a variety by gluing affine pieces) have very concrete interpretations in the toric case, making toric varieties an ideal tool for introducing students to abstruse concepts.
The first half of the class will cover basic material, including affine toric varieties, projective toric varieties, normal toric varieties constructed from fans, divisors, and homogeneous coordinates. The second half will go deeper into the subject, covering topics such as ampleness, cohomology, cotangent bundle and canonical divisor. We'll wrap up everything in the last lecture on toric surfaces.
I will not assume that students have a full background in algebraic geometry; in particular I'll assume no knowledge of schemes, sheaves, or cohomology. However, you should plan to read section zero of each chapter below prior to class. If you spend a couple hours doing the homework (I'll assign 5-6 problems per chapter), you'll get the most benefit out of the class.
Lecture schedule:
Lecture 1: Affine toric varieties
Lecture 2: Projective toric varieties
Lecture 3: The torus action: cones, fans, orbits
Lecture 4: Divisors and sheaves
Lecture 5: Quotient constructions and the Cox ring
Lecture 6: Line bundles and polytopes
Lecture 7: Morphisms and toric bundles
Lecture 8: Cotangent bundle and canonical divisor
Lecture 9: Sheaf cohomology
Lecture 10: Toric surfaces: putting everything together
Grading: Your grade will be determined by homework scores; you'll get out of the class what you put in.
Academic Integrity: I encourage group work on the homework problems. This does not include copying each others solutions.
Textbook: Toric Varieties.
Updated 3/11/25 (hks).