Math 494 Algebraic Geometry

Welcome to Math 494! This course is an introduction to Algebraic Geometry for undergraduates. Algebraic Geometry is the study of solutions of polynomial equations and forms one of the backbones of modern mathematics with applications to Number Theory, Topology and even Logic. Unfortunately, Algebraic Geometry is a broad and technical subject that can be daunting to beginning graduate students. The purpose of this course is to bridge the gap between undergraduate courses and beginning graduate courses in algebraic geometry. We will introduce basic concepts and techniques of algebraic geometry in concrete problems and beautiful examples.

Lecturer: Izzet Coskun, coskun@math.uic.edu

Office hours: M 9:30-11:30, W 9:00--10:00 and by appointment in SEO 423

Time: MWF 1:00--1:50 p.m.

Venue: 317 Taft Hall

Text book: This course will be lecture based. However, there are some nice elementary algebraic geometry texts that I can recommend. The main ones are

• Klaus Hulek, Elementary Algebraic Geometry, AMS Student Mathematical Library Volume 20.
• Miles Reid, Undergraduate Algebraic Geometry, London Mathematical Society Student Texts 12.

Prerequisites: Some knowledge of Algebra (MATH 330), Linear Algebra (MATH 320) and Complex Analysis (MATH 417) is helpful.

Homework: There will be weekly homework. The homework is due on Wednesdays at the beginning of class. Late homework will not be accepted. You are allowed (in fact, encouraged) to discuss problems; however, the write-up must be your own and should reflect your own understanding of the problem. I consider homework to be the most important part of this course. It will account for almost your entire grade. Anyone who misses more than two problem sets will automatically receive a failing grade in this course. Everyone will be required to write a final paper due at the beginning of the final week of the semester. These papers should be 7--10 pages giving an exposition of a theorem not explicitly covered in the course. You should have chosen your topic by the 7th week of classes. The first draft of the paper is due by the 11th week classes.

Grading: Your grade will be based on homework (60%) and a final paper (40%). There will be no in class exams in this course.

Additional references: Here are some other introductory text books in algebraic geometry.

• Karen Smith, et. al., An Invitation to Algebraic Geometry, Springer.
• Brendan Hassett, Introduction to Algebraic Geometry, Cambridge University Press.
• Joe Harris, Algebraic Geometry: A First Course, GTM 133, Springer.

Here are some introductory graduate text books in algebraic geometry. These books are more challenging to read. The purpose of this course is to prepare you to study them. I strongly encourage you to take a look at them from time to time.

• Robin Hartshorne, Algebraic Geometry, GTM 52, Springer.
• Phillip Griffiths and Joe Harris, Principles of Algebraic Geometry, Wiley Classics Library, Wiley Interscience.
• Shafarevich, Basic Algebraic Geometry I, Springer-Verlag.
• David Eisenbud and Joe Harris, The Geometry of Schemes, GTM 197, Springer.

Suggested Paper Topics: Here is a list of suggested paper topics (TOPICS). For your final paper topic, you do not have to choose one of these topics, but in that case you must disucss your choice with me as soon as possible. Please select a topic no later than February 17. You should have a rough draft of your paper by the middle of April (April 21). The final draft is due May 2.

Course materials: