## Combinatorial Algebraic Geometry

## (TCC, Term 1, 2018)

**Lectures**: Thursdays 10:00-12:00 from October 11 to November 29, 2018 (8 weeks)

**Instructor**: Fatemeh Mohammadi

School of Mathematics, University of Bristol

Office: Howard House 5.20

Email: fatemeh dot mohammadi at bristol dot ac dot uk

**Course overview:**

Combinatorial algebraic geometry is the study of varieties with a combinatorial structure. Toric varieties form an important part of this field. Due to combinatorial tools, toric varieties are well-studied and they play an important role in commutative algebra, algebraic geometry and combinatorics. This lecture series will provide an introduction to toric varieties. We will focus more on examples and methods to generate such varieties. There are many great textbooks and lecture notes on toric geometry. I will only present some selected topics on toric geometry following Bernd Sturmfels' book on Gröbner bases and polytopes, and Cox's lecture notes on toric geometry. Here is a rough plan:

**Course organization and scope:**

- Lecture 1: The theory of Gröbner bases and Gröbner fan
- Lecture 2: A crash course in polyhedral geometry
- Lecture 3: Toric ideals
- Lecture 4: Triangulations of polytope
- Lecture 5: Ehrhart polynomial
- Lecture 6: Lattice ideals
- Lecture 7: Toric degenerations of varieties
- Lecture 8: Grassmannians

**Prerequisites for participants:**

- Some familiarity with the basic elements of modern algebra 2 (e.g., fields, groups, rings, ideals and modules) is required.
- The lectures will be self-contained and assume as little background knowledge as possible.
- Students are encouraged to experiment and explore the taught concepts in the computer algebra system Macaulay2 or Singular.

**Registration: **

- Please email graduate.studies@maths.ox.ac.uk with the name of the course and your affiliation.
- Please make sure to fill in this form, too.

**Homework:**

- Eight homeworks (one for each week) will be assigned based on audience background and their interests. Collaboration is both allowed and encouraged, but everyone must write up the solution by himself/herself.
- Homeworks will be posted here after each lecture and they due a week after assignment.
- The solutions should be submitted by email with subject "Week_i Homework" for i from 1 to 8.
- Please either type your solutions or use a marker pen.
- Each homework will be graded and a written feedback will be provided to students.

**Homeworks and Solutions:**

- Weekly homework and a selected set of solutions provided by students will be uploaded here.

**Lecture notes: **

- All lecture notes will be available to students and participants.
- Lecture 1 and Homework 1 (October 11, 2018)
- Lecture 2 and Homework 2 (October 18, 2018)
- Lecture 3 and Homework 3 (October 25, 2018)
- Lecture 4 and Homework 4 (November 1, 2018)
- Lecture 5 and Homework 5 (November 8, 2018)
- Lecture 6 and Homework 6 (November 15, 2018)
- Lecture 7 and Homework 7 (November 22, 2018)
- Lecture 8 and Homework 8 (November 29, 2018) *Homework 8 is at the end of Page 2.*
- Homework 8 is due on December 6, 2018.
- Please make sure to complete the Course Evaluation Survey by December 6, 2018.
- Please make sure to talk to your adminstrative office at your school so that I can submit your marks by December 7, 2018.
- The solutions to Homeworks 1-5 are uploaded here. Please double check the solutions and update/correct yours if needed. When you resubmit them, make sure to add a note which parts have been corrected. The deadline to resubmit them is November 30th.
- Note that each student will receive an individual feedback on the homeworks and the corrections by December 7th, 2018.
- The solutions to Homeworks 1-8 are uploaded here.

**Main references:**

- D. A. Cox, J. Little, and D. O'Shea: Ideals, Varieties, and Algorithms: an introduction to computational algebraic geometry and commutative algebra
- David Cox: Lectures on toric varieties
- R. Stanley: Combinatorics and commutative algebra
- Bernd Sturmfels: Gröbner bases and convex polytopes