MA 726: Algebraic Geometry, NCSU, Spring 2022


Martin Helmer

Email: mhelmer at
Office Hours:


Course notes and assignments will be posted on the Moodle page (click here).


The primary course text will be Invitation to Nonlinear Algebra below. We will refer to this book as MS, the book by Cox, Little and O'Shea as CLO.

Invitation to Nonlinear Algebra by Mateusz Michałek and Bernd Sturmfels.

We will also reference the following books for certain portions of the course:

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

The references below overlap with our course and interested students may enjoy consulting them on occasion. Note, however, that these references contain some material that we will not cover.

Course Goals:

The goal of this course is to introduce students to algebraic geometry in a hands on manner. Students are encouraged to use computer tools such as Macaulay2 or Sage to explore examples and investigate problems.

The primary object at study will by systems of polynomial equations in n variables. The solutions set of a system of polynomial equations forms a geometric object called a variety; we will see that this corresponds to a (radical) ideal in a polynomial ring. We will explore the geometry of varieties both computationally and abstractly using the algebraic structure of polynomial rings.

A major component of this study will be the theory of Gröbner basis. At the end of the course students will be able to answer such questions as: Does a given system of polynomials have finitely many solutions? If so, what are they? If there are infinitely many solutions, how can can these be described and understood?

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.

Suggested Topics: Time line : Mark break down for project: 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).


The Syllabus.

Exam Dates: