Math 176: Algebraic Geometry, Fall 2014

Instructor: Prof. Mohamed Omar
Email: [lastname] at hmc dot edu

Office: Shanahan 2418
Office hours: Monday 5pm-6pm at Shanahan 2418, or by appointment.

Class meetings: Tues/Thurs 9:35 - 10:50 am, Shanahan 2421


Course Outline

This course serves as an introduction to algebraic geometry, one of the most ubiquitous, deepest, and oldest areas of mathematics. The course focuses on the intimate correspondence between curves in space cut out by polynomial equations, and algebraic properties of the ideals defined by these polynomials. Topics will include the study of affine algebraic sets and varieties, local properties of varieties, projective varieties, properties of projective plane curves, and applications of the theory within many different settings.

The course prerequisite is Math 171, and this is an important prerequisite. Most importantly, we will heavily rely on knowledge and experience with commutative rings, especially polynomials rings, so a mastery of this material would be very beneficial for this course. One challenge of an undergraduate algebraic geometry course is having deep material that can be readily explained without an extensive use of mathematical machinery. In light of this, we will be using the book:

''Algebraic Curves'' by William Fulton

as our main source, but this is not required. Class notes should serve as the main reference. The above book is free and available on William Fulton's webpage. A copy of the .pdf will be made available as well.

Course Structure

This course has several graded components. They are:


Lecture Scribing 5%
Presentation 15%
Assignments 20%
Midterm Exam 30%
Final Exam 30%

Lecture Scribing

Each student is responsible for writing lecture notes for at least one lecture in this class. Your grade on this job is based on accuracy and style.

Presentation

Each student will present a 30 minute lecture on an algebraic geometry themed topic of her/his interest. The list of available topics will appear here at a future date. Along with the presentation, each student must create an assignment with 3 problems on it, and write solutions for this assignment. Additionally, a 1 page report on why these problems were selected must be submitted with this material. Each student will have a meeting with Prof. Omar two weeks in advance of their presentation date to go over their plans.

Assignments

Assignments will be due roughly weekly (with the exception of exam weeks) and due Wednesdays at 4:00pm. The assignments will only have a few questions each, and only random segments of each assignment will be graded; this will comprise of 60% of the assignment grade. 20% of the assignment grade will be assigned for completeness. 20% of the assignment grade will be assigned to style points based on solution presentation. There will be no assignments during the presentation weeks.

Midterm Exam

There will be a 75 minute in-class midterm exam on Thurs, Oct 16. Details on what material it will cover are forthcoming.

Final Exam

The final exam will be take-home, 5 hours, and due on Tuesday, Dec 16 at 12:00pm.

Course Policies on Assignments

You are ENCOURAGED to discuss homework problems with other students (unless marked otherwise). Solutions must be written up individually, in your own words. You may use all resources from class. Please note that I know there are easy to find solutions online to almost all of the problems in our textbook. So that you may adequately explore the material in Math 176 on your own, you may not use consult these solutions at any time during the course without my permission. Doing so will be viewed as a violation of the HMC Honor Code.

Disabilities

Students who need disability-related accommodations are encouraged to discuss this with the instructor as soon as possible.

Supplementary Notes

Supplementary notes will be written here.

Midterm Problems

Class Notes

Class notes will be added here.

Date Topic Class Lecture and Author
Sept 2 Affine Algebraic Sets Affine Algebraic Sets Notes Template
Sept 4 Properties of Affine Algebraic Sets Properties of Affine Algebraic Sets (Matthew Wilbur)
Sept 9 Algebra Review Algebra Review
Sept 11 Ideals of Sets Ideals of Sets (Matthew Lin)
Sept 16 Nullstellensatz/Varieties Nullstellensatz/Varieties (Amzi Jeffs)
Sept 18 Irreducible Decompositions of Algebraic Sets Irreducible Decompositions (Nathan Hall)
Sept 23 Polynomial Maps Polynomial Maps (Sarah Armstrong)
Sept 25 Coordinate Ring/Field of Fractions Coordinate Ring/Field of Fractions (Ryan Smith)
Sept 30 Local Rings Local Rings (Abram Sanderson)
Oct 2 Rational Maps Rational Maps (Archer Wheeler)


Assignments

Click here for the preamble file needed to compile the .tex files below.

Here is a great article about writing; good to look at for the ''style'' points: Writing by Prof. Su

Due Date Assignment # .tex version
Sept 10 Assignment 1 Assignment 1 (.tex)
Sept 17 Assignment 2 Assignment 2 (.tex)
Sept 24 Assignment 3 Assignment 3 (.tex)
Oct 1 Assignment 4 Assignment 4 (.tex)
Oct 9 Assignment 5 Assignment 5 (.tex)
Oct 30 Assignment 6 Assignment 6 (.tex)
Nov 13 Assignment 7 Assignment 7 (.tex)
Dec 4 Assignment 8 Assignment 8 (.tex)