Math 143: Elementary Algebraic Geometry

University of California, Berkeley. Fall 2016.

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

Martin Helmer### Book:

** 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.

**Other books/resources:**

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.

- Algebraic geometry: a first course by Joe Harris.
- Algebraic Geometry class notes by Andreas Gathmann.

### 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**

- [Aug. 24] §1.1: Polynomials and Affine Space. M2 Example.
- [Aug. 26] §1.2, 1.3: Affine Varieties, Parametrizations of Affine Varieties
- [Aug. 29] §1.4: Ideals. M2 Example.
- [Aug. 31] §1.5: Polynomials of One Variable. M2 Example.

**Gröbner basis**

- [Sep. 2] §2.1: Introduction to Gröbner basis
- [Sep. 7] §2.2: Orderings of Monomials
- [Sep. 9] §2.3: Division Algorithm
- [Sep. 12] §2.4: Monomial Ideals and Dickson’s Lemma
- [Sep. 14] §2.5: The Hilbert Basis Theorem and Gröbner Bases
- [Sep. 16] §2.6: Properties of Gröbner Bases. M2 Example
- [Sep. 19] §2.7: Buchberger’s Algorithm. M2 Example
- [Sep. 21] §2.8: First Applications of Gröbner Bases. M2 Example
- [Sep. 23] §2.9: Refinements of the Buchberger Criterion. M2 Example

**Elimination Theory**

- [Sep. 26] §3.1: The Elimination and Extension Theorems. M2 Example
- [Sep. 28] §3.1, 3.2: The Elimination and Extension Theorems, The Geometry of Elimination
- [Sep. 30] §3.2: The Geometry of Elimination
- [Oct. 3] §3.3: Implicitization. M2 Example
- [Oct. 5] §3.3, 3.4: Implicitization, Singular Points and Envelopes. M2 Example
- [Oct. 7] §3.4: Singular Points
- [Oct. 10, 12] §3.5: Gröbner Bases and the Extension Theorem

**The Algebra-Geometry Dictionary**

- [Oct. 14] §4.1: Hilbert’s Nullstellensatz
- [Oct. 17] §4.2: Radical Ideals and the Ideal–Variety Correspondence
- [Oct. 19] Midterm
- [Oct. 21] §4.3: Radical Ideals and the Ideal–Variety Correspondence. M2 Example
- [Oct. 24] §4.3: Sums, Products, and Intersections of Ideals
- [Oct. 26] §4.4: Zariski Closures, Ideal Quotients, and Saturations. M2 Example
- [Oct. 28] §4.4: Zariski Closures, Ideal Quotients, and Saturation
- [Oct. 31] §4.5: Irreducible Varieties and Prime Ideals
- [Nov. 2] §4.5: Irreducible Varieties and Prime Ideals
- [Nov. 4] §4.6: Decomposition of a Variety into Irreducibles
- [Nov. 7] §4.8: Primary Decomposition of Ideals
- [Nov. 9] §4.7: Proof of the Closure Theorem . M2 Example
- [Nov. 14] §4.8, §5.1: Proof of the Closure Theorem, Polynomial Mappings

**Polynomial and Rational Functions on a Variety**

- [Nov. 16] §5.1: Polynomial Mappings
- [Nov. 18] §5.3: Algorithmic Computations in Quotients
- [Nov. 21] §5.3, §5.4: Algorithmic Computations in Quotients, The Coordinate Ring of an Affine Variety

**Projective Algebraic Geometry**

- [Nov. 28] §5.4: The Coordinate Ring of an Affine Variety
- [Nov. 30] §8.2: Projective Space and Projective Varieties
- [Dec. 2] §8.2, §8.3: Projective Space and Projective Varieties, The Projective Algebra–Geometry Dictionary

**Review**

- [Dec. 5]: Review Chapter 8 and Chapter 4
- [Dec. 7]: Review Topics

### Assignments:

Assignments and due dates will be posted here. All assignments are due at the beginning of class on the marked due date. All numbers refer to the fourth edition of Cox, Little and O'Shea.- Assignment 1. Submitted September 7.
- §1.1: #6.
- §1.2: #4, #6, one of #10 or #12, #15 (you may use the results from #8-#10 without proof).
- §1.3: #5.
- §1.4: #6, #8, #9, #11, #17.
- §1.5: #8, #9 (you may use a computer algebra system (i.e. M2, Sage, etc) for the gcd's)

- Assignment 2. Submitted September 16.
- §2.2: #2, #11.
- §2.3: #1(a), #9 (you may use a computer for #9(c), but be sure to write what commands you use (i.e. monomial ordering setup, division etc.)).
- §2.4: #7, #8.
- §2.5: #7.

- Assignment 3. Submitted September 26.
- §2.5: #12, #13
- §2.6: #1, #13
- §2.7: #5, #11
- §2.8: #1, #2, #3, #4. (I strongly recommend you use M2 or Sage to compute the Gröbner Bases for these questions, if you do these questions should be short)

- Assignment 4. Submitted October 5.
- §3.1: #1, #2, #5, #6 parts (b) and (c) only,
- §3.2: #3, #4, #5

- Assignment 5. Submitted October 14.
- §3.3: #6, #9, #10, #11
- §3.4: #5, #10
- §3.5: #2, #5

- Assignment 6. Submitted October 26.
- §4.1: #2, #7, #8, #9
- §4.2: #4, #5, #7 (You can use M2/Sage here)
- §4.3: #9, #11, #12

- Assignment 7. Submitted November 9.
- §4.4: #11, #13 (Use M2/Sage for the computation), #14
- §4.5: #4, #7, #11
- §4.6: #4 (Instead of proving by hand just check using M2 or Sage, state what commands you used), #9, #10

- Assignment 8.
**Due December 7**. Total Mark out of 20.- §4.7: #8
- §4.8: #7
- §5.3: #7, #10 (assume that the coefficient field $k$ is an infinite field in §5.3, #10)
- §5.4: #10
- §8.2: #1, #9, #16, #18
**Bonus**(1 additional mark):- §5.4: #9 (uses ideas from Prop. 8 of §5.4 which were not covered in class)

### 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:**

- Any of the the topics listed in Appendix D §2 of our text.
- A topic of your suggestion (please run it by me).
- Automated Geometric Theorem Proving: The goal here would be to expand on what is covered in §6.4 of our text by consulting other references.
- Robotics and Kinematics: The goal here would be to expand on what is covered in §6.1 to §6.3 of our text by consulting other references.
- Invariant Theory of Finite Groups. The goal here would be to summarize and expand on what is found in our text. A starting reference would be Chapter 7 of our text (which we will not cover in class).
- Modern Gröbner Bases algorithms. The goal of this project would be to describe recent advances in algorithms to compute Gröbner Bases. A starting point for this would be Chapter 10 of our text, and in particular §10.3 and §10.4. This project would be a good candidate to include some programming, but it would not be required.
- Numerical Algebraic Geometry: The goal of this project would be to understand how ideas from algebraic geometry can be combined with numerical methods to solve systems of polynomial equations. A starting point for this project could be Introduction to Numerical Algebraic Geometry
- Toric Varieties:
- The projective toric variety associated to a matrix/polytope. One starting reference: Equations Defining Toric Varieties
- A brief introduction to intersection theory and the Chow ring. This is covered in Chapter 1 of the first reference. This project could, for example, focus on a proof of Corollary 2.4 of the first reference below (
**note that this project involves many new concepts**beyond what we cover in class):- David Eisenbud, and Joe Harris. 3264 and all that: A second course in algebraic geometry. Cambridge University Press, 2016.
- William Fulton. Introduction to intersection theory in algebraic geometry. No. 54. American Mathematical Soc., 1984.

**Time line :**

- October 12: Proposal Submission
- November 21: Draft Submission
- November 28: Peer Review Submission
- Dec 7: 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)
- Sage
- Sage Cell
- Sage Math Cloud
- Computations in polynomial rings:

### 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:

- 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.

### 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.### Grading:

Your grades will be broken down as follows:

- Homework: 35%
- Term Project: 28%
- Midterm: 15%
- Final Exam: 22%

### Exam Dates:

- Midterm: Wednesday October 19, in class.
- Final: December 16, 11:30-2:30. (The final exam schedule isn't finalized yet, so this may change)