This semester, Pranav and I are organizing a toric varieties seminar! We will meet Mondays, 3:30-5 in Evans 732. This is a beautiful subject important to a large number of people, including those interested in either algebraic geometry or combinatorics. See below for references and a tentative schedule. The current plan is to do one hour of lecture and then 20 minutes where we'll work an example/exercise. I'll update it as the semester goes along (feel free to bug me if I don't).
Please send me and Pranav (shreepranav_varma@[obvious school].edu) an email to give talks!
Schedule:
September 8: motivation, introduction, the fan construction, examples. Speakers: Joe, Cameron, Pranav
September 15: proof the fan construction works, basic properties of toric varieties. Speaker: Pranav
September 22: how to get the toric variety from the fan, orbit cone correspondence. Speaker: Drew
September 29: divisors on toric varieties, polytopes correspond to ample divisors Speaker: Cameron
October 6: more on the polytope-ample line bundle correspondence, Serre duality and Ehrhart-Macdonald reciprocity, Speaker: Cameron
October 13: Cox ring of a toric variety, GIT construction, Speaker: Lizzie
October 20: Stanley's g-theorem, Speaker: Kaizhe
October 27: Cohomology of line bundles, Bialynicki Birula, Speaker: Cameron
November 3: Volume polynomials, and the Khovanskii-Pukhlikov ring description of the cohomology ring, Alexandrov-Fenchel, Speaker: Xiangru
References:
Fulton's Toric Varieties
Possible Talk Topics: The following topics definitely need to be talked about.
Fan construction: proof of correctness, and basic properties (lots of polyhedral cone geometry involved!)
Orbit cone correspondence and recovering the fan from the toric variety
Polytopes, normal fans and divisors, cohomology of line bundles (might consist of two talks)
Moment map
Below are some ideas I had for interesting talks once we get done with the basics! Any thing related to toric varieties is welcome, however.
Optional Fulton topics:
Topology of toric varieties (fundamental group, Euler characteristics, cohomology)
Resolution of singularities
Intersection theory
Stanley's g-theorem
Bezout's theorem
Serre duality and Ehrhart-Macdonald Reciprocity.
Papers:
Cox Ring: The analogue of the Proj construction for toric varieties. Reference: https://arxiv.org/pdf/alg-geom/9210008
Sumihiro's Theorem: this is the critical fact we need to show that every normal toric variety comes from a fan. Reference: Sumihiro, Hideyasu (1974), "Equivariant completion", J. Math. Kyoto Univ., 14: 1–28, doi:10.1215/kjm/1250523277
Any toric birational map is the composition of a sequence of toric blowups and blowdowns. Reference: R. Morelli, The birational geometry of toric varieties, J. Alg. Geom. 5 (1996), 751–782. MR 99b:14056
GGMS: Stratification of the Grassmanian based on torus orbit closures: Reference: Combinatorial geometries, convex polyhedra and schubert cells, Gelfand, Goresky, Macpherson, Serganova
Batyrev-Blume: describes a general method of getting toric varieties from root systems, the functor of points in this category, and uses this to prove that the permutahedral variety is the Losev-Manin moduli space of stable chains of projective lines with n marked points https://arxiv.org/abs/0911.3607
Richard Stanley: describes a generalization of the h-vector that works for intersection cohomology as well: Generalized H-Vectors, Intersection Cohomology of Toric Varieties, and Related Results
Binomial Ideals: the commutative algebra of the ideals cutting out affine toric varieties (in particular, concerning their primary decompositions): https://arxiv.org/abs/alg-geom/9401001
Michel Brion: Brion's formula: a beautiful formula about polytopes and certain generating functions that can be derived by equivariant K-theory on a toric variety. https://www.numdam.org/articles/10.24033/asens.1572/
Igor Makhlin: One can use Brion's formula to derive the Weyl character formula! https://arxiv.org/abs/1409.7996