Math 143: Elementary Algebraic Geometry

University of California, Berkeley. Fall 2016.

Time: MWF 12-1. Location: 3 Evans.


Martin Helmer
Email: martin.helmer at
Office: 966 Evans
Office Hours: Friday 4-6 PM or see me after class


Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra (Fourth Edition) by David Cox, John Little, and Donal O'Shea. Note that for UC Berkeley students the textbook can be accessed in electronic form on Springer Link via the UCB Library site, link above.

I hope to cover some material (from chapter 10) in the fourth edition that isn't in previous editions, however if you buy a previous edition hard copy you should be fine, just bear in mind that you may need to consult the pdf linked to above from time to time.

Other books/resources:

For those interested: Piazza will be used for Q&A style discussions (click here). To join the class Piazza group click here.

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 the basic principles of algebraic geometry in a hands on manner. Our study will focus on how algebraic methods can be used to answer geometric questions. 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 an 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, this theory will form the basis for our computational approaches to geometric problems. 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? Is so what are they? If there are infinitely many solutions, how can can these be described and understood?

Course Schedule and Notes:

The dates of the lectures are approximate and may be adjusted slightly through the course of the term, in particular I hope to cover more than one section per class for some of the topics in Chapters 2 and 3.

Affine Varieties

Gröbner basis Elimination Theory The Algebra-Geometry Dictionary Polynomial and Rational Functions on a Variety Projective Algebraic Geometry Review


Assignments and due dates will be posted here. All assignments are due at the beginning of class on the marked due date (see Homework Policy below). All numbers refer to the fourth edition of Cox, Little and O'Shea.

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: 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:

Homework will be due once a week (most weeks), day to be determined, at the beginning of class, as a rule late homework will not be accepted. Homework should be handed in on paper in class (this is simpler for the grader). If you are not able to attend class on a given day alternate submission arrangements are possible (such as via email); assignments may also be slid under my office door. Paper submission is preferred, however if you do submit by email please submit .pdf files only (scans are fine). Homework due dates will be posted on this website along with the assignments. Homework assignments will be posted above at least a week before they are due. Each problem set will have a few problems (usually 5-6) that will be handed in, of these 2-3 will be graded. There will also be a longer list of practice problems. The material on the practice problems will be covered on quizzes and exams.

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.

Incomplete Grade Policy:

Per University policy an "incomplete" grade will be granted only in cases where a student has completed more than 75% of the term work, and is receiving a passing grade on this work, but is unable to complete the course due to documented circumstances beyond their control.


Your grades will be broken down as follows:

Exam Dates: