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

ANU. Semester 1, 2021.

Lecture: Wednesday, 4:00 pm -- 6:00 pm.

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

Location: Usually HN 1.37, see weekly schedule below for exceptions to the usual time and location. Zoom Attendance Possible (click here).

### Instructor:

### Books:

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].

**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.

##### Prerequisites:

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**

- Week 1 (March 3, HN 1.37):
*Informal Definition of a Toric Variety and a Brief Review of Algebraic Geometry*.

**Embedded Toric Varieties (Toric Varieties and Polytopes)**

- Week 2 (March 10, HN 1.37):
*Affine Toric Varieties and Toric Ideals (§1 of [S])*

- Week 3 (Workshop March 15 in HN 2.42. Lecture March 17 in HN 1.37):
*Affine Toric Varieties and Toric Ideals (§1 of [S])*

- Week 4 (Workshop March 22 in HN 1.37. Lecture March 24 in HN 2.48):
*Projective Toric Varieties and Polytopes (§2.1 of [S])* - Week 5 (Workshop March 29 in HN 1.37. Lecture March 31 in HN 1.37.):
*Kushnirenko's Theorem (§2.2 of [S])* - Week 6 (Workshop April 19 in HN 1.37. Lecture April 21 in HN 1.37.):
*Kushnirenko's Theorem (§2.2 of [S])*

**Toric Varieties from Fans and Abstract Toric Varieties**

- Week 7 (No Workshop. Lecture April 28 in HN 2.48.):
*Cones and Fans (§3.1 of [S])*

- Week 8 (Workshop May 3 in HN1.37. Lecture May 5 in HN 1.37.):
*Toric Varieties from Fans (§3.2 of [S])*

- Week 9 (Workshop May 10 in HN1.37. Lecture May 12 in 2.42, starting at 4:30 pm):
*Toric Varieties from Fans (§3.2 of [S] §3.3 of [S])*

**Homogeneous Coordinates on Toric Varieties (the Cox Ring) and/or Mixed Volumes and Bernstein's Theorem (§4 of [S])**

- Week 10 (Workshop May 17 in HN1.37. Lecture May 19 in HN1.37):
*Homogeneous Coordinates on Toric Varieties (Chapter 5 of [CLS])*

- Week 11 (Workshop May 24 in HN1.37. Lecture May 26 in HN 2.48 (the MSI Boardroom)):
*Homogeneous Coordinates on Toric Varieties (Chapter 5 of [CLS])*

### Assignments:

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].

- Assignment 1. Due April 2.
- Regular Problems: #1 on Pg.7 of [S]. #2 on Pg.7 of [S]. #9 on Pg.8 of [S]. #3 on Pg.129 of [MS]. #1 on Pg.18 of [S]
- Programming/Computer Problems:

1) Write a program which takes as input a full rank matrix with positive integer entries and outputs the associated toric ideal in the appropriate polynomial ring over the rationals.

2) With the aid of your software program complete #3 and #4 of [S].

- Assignment 2. Due April 30.
- Regular Problems: #2, #3, #4, #5, #6 on Pg 18 of [S]
- Programming/Computer Problems: #11 on Pg 19 of [S]

- Assignment 3. Due May 20.
- Regular Problems: #10 on Pg 18 of [S]. #13 on Pg 19 of [S] (you may consult the proofs given in Chapter 8 of [MS] and in [CLS], but should write your own version). #5, #6, #9 on Pg 30 of [S].
- Programming Problems: Assignment 3 Programming Questions.

- Assignment 4. Due June 11.
- Regular Problems: #7 on Pg 30 of [S], #5.0.1, #5.0.3, #5.2.2 of [CLS]
- Programming Problems: Assignment 4 Programming Questions.

### 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 Z.Z. will present #8 on Pg 18 of [S].

May 10: T.B. will present #1 on Pg 30 of [S].

May 17: Y.Y. will present #3 (using #2 as a fact if needed) on Pg 30 of [S].

May 24: C.D. will present #4 on Pg 30 of [S].

### 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:**

- A topic of your suggestion (please run it by me).
- Tropical Geometry: Another area where combinatorics and geometry interact, see the book "Introduction to Tropical Geometry" by Bernd Sturmfels and Diane Maclagan.
- Sheaf Cohomology of Toric Varieties: See Chapter 9 of [CLS].
- Toric GIT and the Secondary Fan: See Chapter 14 of [CLS].
- Gröbner Fans: See the book "Gröbner Bases and Convex Polytopes" by Bernd Sturmfels and references therein. For an algorithmic project consider the paper: 'Computing Gröbner Fans of Toric Ideals' by Birkett Huber & Rekha R. Thomas
- Toric Resolutions and Toric Singularities: Chapter 11 of [CLS].
- The Topology of Toric Varieties: See Chapter 12 of [CLS].
- Toric Dynamical Systems and Chemical Reaction Networks: See the article "Toric Dynamical Systems" by G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels.
- Discrete Exponential Models in Algebraic Statistics: See the first three chapters of Lectures on Algebraic Statistics by Mathias Drton, Bernd Sturmfels, and Seth Sullivant. Also see the book "Algebraic Statistics" by Seth Sullivant.
- The Canonical Divisor of a Toric Variety: See Chapter 8 of [CLS].
- Toric Ideals and Phylogenetics: See the article "Toric ideals of phylogenetic invariants" by B. Sturmfels and S. Sullivant.
- Hypergeometric Functions and Toric Varieties: See the paper "Hypergeometric Functions, Toric Varieties and Newton Polyhedra" or the book "Discriminants, Resultants, and Multidimensional Determinants", both by Gelfand, Kapranov, and Zelevinsky.
- Algorithms for toric ideals, e.g. Gröbner Bases of toric ideals. Some references: A.M. Bigatti and R. LaScala and L. Robbiano. Computing toric ideals. Journal of Symbolic Computation 27 (1999), 351--365; R. Hemmecke and P. Malkin. Computing generating sets of lattice ideals; S. Hosten and B. Sturmfels. GRIN: An implementation of Gröbner bases for integer programming; P. Malkin. Truncated Markov bases and Gröbner bases for Integer Programming, Chapter 11 in: J. A. De Loera, R. Hemmecke, M. Köppe. Algebraic and Geometric Ideas in the Theory of Discrete Optimization.
- Singularities of Toric Varieties, A-resultants, A-discriminants, A-determinants and how you compute them: A starting reference is the notes (and the references therein): A brief survey of A-resultants, A-discriminants, and A-determinants by Madeline Brandt, Aaron Brookner, and Christopher Eur.

**Time line :**

- April 7: Proposal Submission
- June 3: Draft Submission
- June 10: Peer Review Submission
- June 17: Project Submission

**Mark break down for project:**

- Main report: 80% of your project grade
- This should be an approximately 10 page report detailing what you have studied. It should give the key definitions and build to some sort of main result, either a theorem or algorithm, and it should detail the theoretical basis for this result in either case. Your report should be written so that other students in the class can understand it.
- Peer review: 10% of your project grade
- As part of your projects you will be asked to swap a first draft with one other student in the class. You will then write feedback on the other students report, this feedback should point out any mathematical (or typographical) errors you note and should offer comments on things that could be clarified. Your mark for this will be based on your comments, i.e. on how well you critically read the other students work
- Proposal: 10% of your project grade
- A one page outline of what you propose to study and how you plan to do so.
- There may be an option to give a talk on your project, but this will depend on scheduling and level of interest.

**Group Work:**

- You may partner with another student to jointly explore a related topic, however you must each submit your own independently written term paper. In particular you and your exploration partner should focus on at least slightly different aspects of the topic.
- You may have overlapping (but independently written) definitions and background, but your main result should be different.
- If you find these guidelines unclear please speak to me about it.

**LaTex Example File**

- Your project (and ideally your proposal) should be prepared using LaTex. To get you started there is an example file below.
- The tex file
- The resulting pdf 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).- Macaulay2
- Download and install
- Use online
- M2 may also be used online via the Sage Math Cloud. To use M2 this way create an account (this is free) and open a new project, then start a new Terminal (>_ icon). Once the Sage terminal opens type the command: M2. M2 will then open and you can enter M2 commands.
- Computations in polynomial rings:
- Polynomial Rings
- Quotient Rings
- Factoring Polynomials
- Computing gcd
- Division with remainder, i.e. getting a representative in a quotient ring, inverting an element in a quotient ring which is also a field etc.
- Check if an integer or ideal is prime
- Check equality of ideals (or numbers or other objects)
- Download/install Ubuntu 18.04 from the Windows Store
- Follow the Ubuntu Install instructions for M2 (if you use 18.04 Ubuntu the following should do it)
`sudo echo 'deb http://www.math.uiuc.edu/Macaulay2/Repositories/Ubuntu bionic main' >/etc/apt/sources.list.d/macaulay2.list`

`sudo apt-key adv --keyserver hkp://keys.gnupg.net --recv-key CD9C0E09B0C780943A1AD85553F8BD99F40DCB31`

`sudo apt-get update -q`

`sudo apt-get install -y -q macaulay2`

- Install Xserver and emacs (if desired)
- Sage
- Sage Cell
- Sage Math Cloud
- Computations in polynomial rings:

#### Installing M2 in Windows

### 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:

- 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. - 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. - 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**. - If the grader is unable to make out your writing then this may hurt your mark.

### Grading:

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:

- Assignments: 44%
- Written Portion of Term Project: 42%
- Oral Presentation on Term Project: 7%
- Participation in Workshop Discussions: 7%

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

- Assignments: 45%
- Term Project: 45%
- Participation in Workshop Discussions: 10%