Introduction to Algebraic Geometry (Math 8620), Fall 2014
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
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!
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.
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!
- Algebraic Geometry,
by Robin Hartshorne (The (abridged) Bible. Great source of
exercises. More difficult than the Red Book and contains more
material. A must for those seriously interested in algebraic geometry.)
- Éléments de Géométrie Algébrique,
by Alexander Grothendieck (The unabridged Bible. 1800 pages over 4
volumes, but very logical and proves everything in complete
detail. Also includes the necessary commutative algebra.)
- Foundations of Algebraic Geometry, by Ravi Vakil (lecture notes, more abstract from the beginning. Does not start with varieties.)
- The Geometry of Schemes, by David Eisenbud and Joe Harris (very geometric and example-based)
- Algebraic Geometry and Arithmetic Curves, by Qing Liu (lots of material related to number theory and my research!)
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
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.
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.
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