Ma132C - Toric geometry (Spring 2025)

InstructorYassine El Maazouz
Course webpage http://www.yelmaazouz.org/Ma132C/index.html
Course Code Ma 132 bc
LecturesTu-Th 13:00–14:30, 255 Linde Hall
E-Mail:maazouz [at] caltech [dot] edu
Office308 Linde Hall of Mathematics and Physics
Office HoursTh 15:00–16:00
Excluding holidays. Please email me if you cannot make any of these times.
Prerequisites Ma130ab is strongly recommended. I will assume familiarity with basic algebraic geometry (chapters I and II in Hartshorne's book).
Required Text

There is no required text for this course. However, the following can be used as references:
  1. [F] William Fulton. Introduction to toric varieties. Annals of Mathematics Studies, 131. Princeton University Press, Princeton, NJ, 1993. .

  2. [CLS] David Cox, John Little and Henry Schenck. Toric varieties. Graduate Studies in Mathematics, 124. American Mathematical Society, Providence, RI, 2011 .

Catalog Description
This course will cover advanced topics in algebraic geometry that will vary from year to year. Topics will be listed on the math option website prior to the start of classes. Previous topics have included geometric invariant theory, moduli of curves, logarithmic geometry, Hodge theory, and toric varieties. This course can be repeated for credit. Part a and part b not offered 2024-25.

Syllabus
I plan to cover the following topics (depending on how much progress we make):
  1. Affine and projective toric varieties.
  2. Normal toric varieties from rational polyhedral fans.
  3. Divisors and line bundles on toric varieties.
  4. Sheaf cohomology of toric varieties.
  5. GIT quotients and toric varieties.
The goal of the course is to understand the basic theory of toric varieties and the interplay between their geometry and combinatorics.

Homework
Bi-weekly problem sets will be assigned. While collaboration among students is encouraged, you must ensure you fully understand the solutions and write them independently. I strongly recommend attempting the problems on your own before seeking help or discussing them with others. Additionally, you are required to list all collaborators and sources consulted for each assignment.


Presentations
Students enrolled in this course for credit must deliver a presentation on a topic related to toric varieties. A list of suggested topics will be provided soon, but you are also welcome to choose your own topic, provided it aligns with the course material. The duration of each presentation will be determined by the number of participants. Presentations must be given without the use of notes.

Grading The final grade will be based 60% on weekly problem sets and 40% on the presentation.