Prerequisites: Math 248A, but please contact me if you are
interested and have not taken this course
Textbook: Hartshorne, Algebraic Geometry
Syllabus: We will cover basic aspects of complex algebraic
varieties, and schemes. See also the
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 begin with a discussion of complex varieties, and then
move on to schemes. The schemes material will be largely self-contained,
but the material on complex varieties will assume some background from
248A; the most important concepts from the first
quarter will be abstract varieties, nonsingular varieties, and complete
I shall, from time to time, post lecture notes on topics supplemental
to what is covered by Hartshorne.
Complex varieties and the
analytic topology: we define the analytic topology on complex
varieties, and explore its basic properties in relation to the properties
of algebraic varieties which we have already defined.
Power series and nonsingular
points: we define the completion and power series expansions at
nonsingular points, and use this to prove basic properties of nonsingular
points of varieties.
Analytic functions and
nonsingularity: we use atlases to define analytic spaces, and show
that a complex prevariety yields a complex manifold if and only if it
is nonsingular. We also describe some applications of complex techniques.
Problem sets will be posted here each Thursday, due the following Thursday
in class. You are encouraged to collaborate with other students, as
long as you do not simply copy their answers.
Problem set #1, due 1/14: Do Exercises 2.2 and 2.6 of the
Complex varieties and the analytic topology lecture notes (these
should be quite straightforward, but note that the ordering of parts in
2.2 is more to have a convenient statement of results than to suggest
the order of proof, so you may find it more efficient to use a different
order/structure for your arguments). Also show that if X is a
compact topological space, then for every topological space Y the
projection map X×Y→Y is closed.
Problem set #2, due 1/21: Do Exercise 4.3 of the
Power series and nonsingular points lecture notes.
Problem set #3, due 1/28: Do Exercise 2.2 of the
Analytic functions and nonsingularity lecture notes.
Problem set #4, due 2/4: Do Exercise 1.21 of Chapter II
Problem set #5, due 2/11: Do Exercises 2.2 and 2.9 of Chapter II
Problem set #6, due 2/18: Do Exercises 3.9 and 3.10 of
Chapter II of Hartshorne.
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.
The paper is due on Tuesday, 3/16.
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.
The paper should be roughly 8-10 pages in default tex formatting. Using
latex with the amsart documentclass is recommended, but if you are
already familiar with another flavor of tex, you may use that instead.
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
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.