MATH3349/MATH4349/MATH6209 (Special Topics in Math): Toric Varieties and Combinatorial Methods in Algebraic Geometry

ANU. Semester 1, 2021.

Lecture: Wednesday, 3:30 pm -- 5:30 pm.

Workshop: Monday, 2:00 pm -- 3:00 pm. Typically 40 minutes long.

Location: HN 2.42. Zoom Attendance Possible (click here).


Martin Helmer

Email: martin.helmer at
Office: 4.70, Hanna Neumann Building #145
Office Hours: Monday 3:30 pm -- 4:30 pm or by appointment.


The lectures will feature material from the lecture notes of Sottile, the textbook of Cox, Little, and Schenck, and the textbook of Fulton, with the notes of Sottile being the primary reference.

Lectures on Toric Varieties by Frank Sottile. Denoted as [S].

Toric Varieties by David A. Cox, John B. Little, and Henry K. Schenck. Denoted as [CLS].

Introduction to Toric Varieties by William Fulton. Denoted as [F].

For students wishing to review concepts more generally from algebraic geometry some suggested books are below:

Invitation to Nonlinear Algebra by Mateusz Michałek and Bernd Sturmfels. Denoted as [MS].

Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra (Fourth Edition) by David Cox, John Little, and Donal O'Shea.

Course Goals:

Toric varieties are a large class of algebraic varieties which are naturally understood using geometric combinatorics and often serve as important test cases when exploring properties of algebraic varieties in general. The goal of this course is to introduce students to toric varieties and to explore how one can use combinatorial techniques to understand their geometric and algebraic properties. There will be an emphasis on obtaining a hands-on understanding through examples and computations. Students are encouraged to use computer tools such as Macaulay2 or Sage to explore examples and investigate problems.


A first course in abstract algebra (e.g. Algebra 1) is required. Some familiarity with algebraic geometry is recommended, partly to understand the context of the topic, but is not strictly required. The relevant definitions/theorems from algebraic geometry will be reviewed (without proof) either in the review lecture at the beginning, or as needed throughout the course.

Course Schedule and Notes:

The pace of the lectures is approximate and may be adjusted slightly through the course of the term; the topics covered in lectures 11 and 12 may also be modified at a later date. Hand written notes from lecture will be uploaded here in the corresponding place in the schedule.

Introduction to Toric Varieties and Review of Algebraic Geometry

Embedded Toric Varieties (Toric Varieties and Polytopes) Toric Varieties from Fans and Abstract Toric Varieties Real Toric Varieties Homogeneous Coordinates on Toric Varieties (the Cox Ring) and/or Mixed Volumes and Bernstein's Theorem (§4 of [S])


Assignments and due dates will be posted here. All assignments are due at the beginning of class on the marked due date. The questions will primarily be taken from exercises in [S].

Workshop Presentation Schedule:

Each week in workshop one student will present a problem to the class, this forms part of your workshop participation grade. Your assigned problem will be posted here at least a week in advance. The presentation assignments vary in difficulty, this will be taken into account on grading. Further, student presenters will primarily be graded on their presentation; i.e. even if you get stuck on your assigned problem you can still get a good grade, just clearly present what you understood and what the challenges are.

March 22: L.M. will present #5 on Pg 7 of [S].
March 29: L.C. will present #7 on Pg 7 of [S] (Note this makes use of Buchberger's Algorithm, I will give an overview of this in the Workshop on March 15).
April 19: Z.D. will present #7 on Pg 18 of [S].
May 3 [Two Presentations by Z.Z. and T.B.]: Z.Z. will present #8 on Pg 18 of [S]. T.B. will present #7 on Pg 30 of [S].
May 10: G.L. will present #9 on Pg 30 of [S].
May 17: Y.Y. will present...
May 24 [Two Presentations by C.D. and E.P.]: C.D. will present... E.P. will present...

Term Project/Paper:

This course will involve a term project. The project will require students to independently study a class-related topic. The results of your work and the understanding that you have gained will be summarized in a short paper. Your paper should be self-contained and should be written so that to the other students in our class can understand it. The target length will be approximately 10 pages. If appropriate your project may also have a software component, in such cases the report may be somewhat shorter but should still contain the ideas behind the algorithms present in your software. If you are unsure about what the topics below entail please feel free to ask me for more details, I am happy to give an informal overview of the area and the ideas involved.

Suggested Topics: Time line : Mark break down for project: Group Work: LaTex Example File

Algebra Software:

Macaulay2 (M2 for short) and Sage are both excellent open source computer algebra systems with some very helpful functions for algebra, algebraic geometry and number theory (among other things).

Homework Policy:

There will be four assignments, worth 11% each. These will consist primarily of problems from the course texts/notes.

Some things to keep in mind when doing your homework:

  1. You are encouraged to discuss problems with your classmates and are free to consult online resources. Working together on math problems can be an excellent way to learn and the internet is a useful resource. However your final written solutions you hand in must be your own work written in your own words, that is your final solutions must be written by yourself without consulting someone else's solution.
  2. All solutions should be written in complete, grammatically correct, English (or at least a very close approximation of this) with mathematical symbols and equations interspersed as appropriate. These solutions should carefully explain the logic of your approach.
  3. All proofs must be complete and detailed for full marks. Avoid the use of phrases such as 'it is easy to see' or 'the rest is straightforward', you will likely be docked marks. Proofs in your homework should be clear and explicit and should be more detailed than textbook proofs.
  4. If the grader is unable to make out your writing then this may hurt your mark.


You will have the option to give a final talk on your term project; this will effect the grade breakdown. Giving a final talk is encouraged, but not required.

If you do give a presentation the grades will be broken down as follows:

If you do not give a presentation the grades will be broken down as follows: