Meeting: Room 2240 MSB from 9:00 to 10:30 Tuesdays for presentations and
room 2112 MSB from 10:00 to 11:00 Thursdays for problem sessions.
Credit: The variable unit request form can be found here https://www.math.ucdavis.edu/grad/gr ad-student-handbook and should be filled out if you want course credit.

References:
[CLS] Toric Varieties by D Cox, J Little and H Schenck.
[K] Equivariant Bundles on Toral Varieties by A Klyachko.
[DJS] Toric Vector Bundles and Parliaments of Polytopes by S DiRocco, K Jabbusch and G Smith.



This term will be accessible to students with some algebraic geometry background. The previous term of toric varieties will be helpful and motivational but not needed. The main reading will be chapters 4, 6 and 7.3 from the book by Cox, Little and Schenck which cover divisors, numerical effectiveness with line bundles (nef and Mori cones) and a classification theorem of Kleinschmidt which uses these techniques respectively. The fundamental idea is that many toric varieties arise from the fiberwise projectivization of split vector bundles over simpler ones and Kleinschmidt's result is a situation in which this construction gives all toric varieties with some properties. The proof allows for practice with Cartier divisors and nef cones in a setting in which they can be very explicitly studied. Many of the technical details are set as exercises so the methods should become familiar. Relevant exercises include 6.1.5, 6.2.2, 6.3.4, 7.3.4 and 7.3.6 from [CLS]. The meetings will be scheduled for 90 min each week and will be presentations by students. The material is about equally divided between proofs and examples such as weighted projective spaces and Hirzebruch surfaces. This will likely take most of the term but there are interesting directions to go from there. One is the paper of Klyachko and the combinatorial notion of parliaments which look at nonsplit vector bundles over toric varieties. Another is studying other cohomology theories such as equivariant and Chow for toric varieties.

zoom:Available upon request: 715 058 8313

Seminar Outline:

Day	Speaker	Topic	Exercises
January 07
January 14	Timothy Paczinski	Divisors: Chapter 4.0 of [CLS]	4.0.12, 4.0.13
January 21	Daniel Qin	Toric Divisors: Chapter 4.1/2 of [CLS]	4.1.3, 4.2.1, (4.2.10)
January 28	Jake Garcia	Sheaves and Torus Invariant Divisors: Chapter 4.3 [CLS]
February 04	Evan Ortiz	Sheaves to Line Bundles: Chapter 6.0 of [CLS]
February 11
February 18
February 25
March 04
March 11