Math 666: Toric varieties

Instructor: Frank Sottile
Lectures: TuTh 9:35–10:50 Blocker 306
Course webpage: https://franksottile.github.io/teaching/24.2/Toric.html
Grading: Based on class participation and end of term project.
Office Hours: Blocker 601L Mondays 14:00-15:00
Tuesdays 11:00-12:00
About this course

Among the most accessible classes of algebraic varieties are toric varieties. This is fortunate for these are also among the most commonly encountered outside of algebraic geometry within mathematics and in the applications of algebraic geometry. Toric varieties in particular are currently widely studied in algebraic geometry and its applications.

This course would introduce the students to toric varieties, emphasizing their fundamental combinatorial nature while focusing on concrete examples and explaining some of their applications.

Because of the elementary nature of these varieties, the prerequisite will be graduate algebra, although courses in commutative algebra and algebraic geometry will be helpful.

Grading

I feel that in advanced graduate classes, the students get out of the class what they put in. Consequently, I do not assign regular homework to be collected and marked. (But I will mention exercises, which are intended for you to fill in gaps in the presentation, thereby learning a bit more.) At the end of the term, students will present projects. <-- Here is a file with some suggestions.-->
While I do not collect written work, here are some exercises.
Here is information about the presentations.

Macaulay2 Scripts Topics:
  • Toric ideals
    • Generation of toric ideals
    • Groebner bases
    • Computation
  • Affine toric varieties
  • Projective toric varieties
    • Orbit decomposition and Limiting behavior
  • Lattice Polytopes
    • Algebraic-Combinatorial Geometric Dictionary
    • Normality of Sturmfellian Toric Varieties
  • Real projective toric varieties
    • Irrational toric varieties
    • Algebraic moment map
  • Geometric combinatorics, lattices and fans
  • Abstract Construction of Toric Varieties
  • Cox/Delzant quotient construction
    • Cox Coordinate Ring
  • Line bundles on toric varieties
  • Toric Degenerations
  Text Book:
    I do not plan to follow a text book for this class. I will be using my own notes.
Some Sources:
I have some notes on algebraicc geometry and Gröbner bases; these are the first two chapters in the following set of notes (the others are unfinished). Algebraic Geometry for Applications.
I have some notes that I wrote for a CIMPA school in Ibadan, Nigeria in 2017: Ibadan Lectures on Toric Varieties.
I can recommend the monumental book Toric Varieties by Cox, Little, and Schenck. This is an encyclopedia of toric varieties, with a wealth of material, far more than can be digested in a term. This is and will be the definitive source for the subject for many years to come. It is published by the American Mathematical Society
Another popular book is Fulton's Introduction to Toric Varieties, by Princeton University Press. Its publication in the 1990's made the subject accessible and really opened up the field. It is not as elementary as the book by Cox, Little, and Schenck or that by Ewald, but it is much shorter and gives a good introduction to the subject.
This book by Ewald treats both the algebraic geometry of toric varieties, and the related geometric combinatorics. Many from combinatorics find it the right introduction to the subject.

Last modified: Sat Nov 09 10:31:44 CST 2024 by sottile