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 the basic theory of schemes, with some
material on complex varieties. 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 continue with the material from 248A, starting by reviewing
and elaborating on the definition of schemes. We will also discuss basic
properties of schemes, and how they relate to the classical theory
of complex varieties.
I shall occasionally post lecture notes on topics supplemental
to what is covered by Hartshorne.
Valuative criteria: Gives
statements and proofs of the valuative criteria for separatedness,
universal closedness, and properness.
Properties of fibers and
applications: we study irreducibility and reducedness of schemes
over fields, and the behavior under extension of base field. We examine
how these ideas relate to fibers of morphisms, and finally apply the
results to prove that any two (closed) points of a variety can be connected
by a (not necessarily irreducible) connected curve.
Complex varieties and the
analytic topology: We introduce the analytic topology on a scheme
of finite type over the complex numbers, 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
Problem sets will be posted here each Monday, due the following Monday
in class. You are encouraged to collaborate with other students, as
long as you do not simply copy their answers.
Problem set #1, due 1/18 (1/16 is a holiday): read sections 1
and 2 of Chapter II of Hartshorne, excepting the material on Proj of a
graded ring. Pay special attention to parts of the presentation which are
different from Mumford. Choose two exercises from each section that aren't
immediately obvious from 248A, and do them.
Some comments for problem set #1.
Problem set #5, due 2/13:
(1) Show that if X is a
reduced scheme, Y any scheme, and Z a subscheme of Y,
then a morphism f from X to Y factors through Z
if and only if its image is contained in the underlying set of Z
(note that this is more or less equivalent to the last part of Hartshorne's
exercise 3.11 (d), so you may not cite this).
(2) Show that if X and Y are schemes over Spec k for
some field k, the points X(k) of X with residue
field k are dense in X, X is reduced, and Y is
separated over Spec k, then two morphisms from X to Y
over Spec k which agree (set-theoretically) on X(k) are
in fact the same morphism. Conclude that even if Y is not separated,
given the hypotheses on X if two morphisms agree on the underlying
sets then they agree.
(3) Using either (1) or (2), follow the argument sketched in class to conclude
that if X is reduced, connected, universally closed and of finite
type over an algebraically closed field k, then the global sections
of the structure sheaf of X are precisely k.
Selected solutions for problem set #5.
Problem set #6, due 2/24: In Chapter II of Hartshorne, read
the definition of finite morphism, then do exercises 3.5, 3.8, 3.20, and 4.1.
For 3.8 and 3.20, work directly from what's in Hartshorne, without using
further statements from class.
Selected solutions for problem set #6.
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, 3/20.
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.