Syllabus: We will cover the basic theory of schemes, with some
material on complex varieties. See also the
department syllabus,
although we will begin with the schemes material

Grading: 75% homework, 25% final expository paper

Homework: Homework will be assigned roughly weekly until the
last few weeks of the quarter, when you will begin to work on your paper.

Welcome to Math 248B: Algebraic Geometry

Algebraic geometry is the study of solutions of systems of polynomials
equations. It is a classical field with a long history, which
has a close relationship to many fields of pure math, but has also recently
been applied to areas as diverse as engineering, computer
graphics, cryptography, and algebraic statistics, to name a few.
In 248B, we will start by introducing schemes, and studying
their basic properties. We also give a brief presentation of the
classical topology on complex varieties.

Lecture notes

I shall occasionally post lecture notes on topics supplemental
to what is covered by Hartshorne.

Moduli spaces, representable
functors, and fibered products: discusses representable functors
in the context of moduli spaces and universal properties. Introduces
Zariski sheaves, gives a criterion for representability, and uses it
to prove existence of the fibered product.

Separated and proper
morphisms: discusses separatedness and properness, including
the valuative criterion. Also briefly discusses properties of properties
of morphisms.

Complex varieties and the
analytic topology: We introduce the analytic topology on complex
prevarieties, and show that separatedness, properness and connectedness
are equivalent to Hausdorffness, compactness, and connectedness in the
analytic topology. In the process, we also show that a nonsingular complex
variety has the natural structure of a complex manifold.

Problem sets

Problem sets will be posted here each Wednesday, due the following Wednesday
in class. You are encouraged to collaborate with other students, as
long as you do not simply copy their answers.

A final paper will be due at the end of the quarter. This will be an
expository paper on a supplemental topic of your choice.

Guidelines:

The paper is due on Monday, 3/17.

Please come to my office hours as soon as possible to discuss
your choice of topic. It should be narrow enough that you can give at
least one result statement and some argument. Reasonable topics could
include deformation theory, toric varieties, group schemes, moduli
spaces, degeneration arguments and intersection theory, but there are
many other possibilities.

The paper should be roughly 8-10 pages in default tex formatting. Using
latex with the amsart documentclass is recommended.

There is no requirement (nor encouragement) of originality of content,
but of course the paper should be entirely in your own words.

The paper should be fully self contained.
In particular, it should have a brief introduction explaining the
main material to be discussed, and ideally some explanation of why it is
important.

Material from lecture may be used without citation. Material
not covered in lecture but which is in Hartshorne may be used with a
citation without explicitly reproducing the statement. Any other
material should be used with a precise statement of the result needed as
well as a citation.