Introduction to Algebraic Geometry (Math 8620), Fall 2014  

 
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
phone: 434-424-4930
website: http://people.virginia.edu/~aso9t/math8620f14.html

 

Lectures

MWF 11:00-11:50, Kerchof Hall 317.  Please ask questions if anything in lecture is unclear. Lectures will run the entire 50 minutes. Please show up on time!



Problem session

There will be an (almost) weekly one-hour problem session where students will present homework solutions on the board, and where we can discuss the solutions.  The problem session will meet Fridays from 9-10 AM in Kerchof 326.


Textbook

The Red Book of Varieties and Schemes, by David Mumford.  There should also be a copy on reserve in the library.  This book has great exposition and a lot of geomteric intuition, while still taking a very algebraic approach.  It also has some nonstandard terminology/notation and unfortunately more than its share of typos.  I'll try to point both of these out as much as possible!

Other references for more-or-less standard material include:
If you want references for more advanced topics (moduli spaces, elliptic curves/abelian varieties, intersection theory, étale cohomology, arithmetic geometry, group schemes, transcendental methods...), or for algebraic geometry on a more basic level, please talk to me!

Content

Algebraic geometry is the study of spaces of solutions of polynomial equations, in some very general sense.  It has evolved from the analytic geometry of Descartes to its state today, where it has close links with algebraic topology, number theory, complex analysis, commutative algebra, Galois theory, representation theory, and category theory.  The subject underwent a revolution in the 1960s due to Grothendieck's introduction of the concept of scheme.  Schemes are general enough to encompass everything from closed complex manifolds in projective space, to rings of integers of number fields, to p-adic disks, and are useful even for answering questions that come from the simpler world of varieties (roughly, spaces that locally look like the set of solutions to a system of polynomial equations over a field). Since the intuition for schemes comes from varieties, we will spend the first month or so of the course discussing varieties, but we will move on to schemes as quickly as possible.

In particular, I hope to cover chapters I and II of the Red Book (some of which you already know from Algebra IV).  If there is time left over, I might do a small unit on divisors or on differentials.  One semester is not a lot of time for algebraic geometry, and I've had to make some decisions about what to exclude.  Unfortunately, there will not be a great "capstone theorem" for this course --- the goal is more to get you conversant and comfortable with the language of schemes and to give you intuition.  If there is demand, I will run a reading course in the Spring to cover further topics.

The prerequisite for this course is knowledge of commutative algebra at the level of Atiyah-MacDonald (including many of the exercises there).  In particular, you should be comfortable with the idea of a ring homomorphism A --> B giving rise to a continuous map Spec B --> Spec A of topological spaces.  If you need to brush up, Chapters 1, 2, 3, 5, 7, 9, and 10 are the most important for this course.

Office Hours

Mondays 1-2, Fridays 12-1.  Kerchof Hall 208 (my office). If these times do not work for you, please make an appointment with me.


Homework

Homework will not be handed in, but you should be prepared to discuss/present solutions at the problem session, which will meet most weeks.  You are encouraged to work together on the homework, although I have found that this is more effective when you have first spent some time thinking about the problems by yourself.  It is a good idea to TeX up full solutions (I still reference my grad school homework solutions every once in a while).  If you want me to look over an answer, I am happy to.

There will be roughly 6-7 homework assignments over the course of the semester.  Some of the problems will be difficult!  The homework assignments will be posted on Collab.


Exams

There are no exams in this class.


Comments

If you have (anonymous) comments for me about teaching style or anything related to the course, you can make them on the Collab page for the course.