## Math 248B - Algebraic Geometry Winter 2012

Instructor: Brian Osserman

Lectures: MWF 3:10-4:00, Physics 130.

CRN: 54109

Office: MSB 3218, e-mail:

Office Hours: W 4:10-5:00, F 2:10-3:00

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 department syllabus.

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.

### Lecture notes

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

• Points and morphisms, from the classical and scheme-theoretic points of view: discusses affine algebraic sets over an arbitrary field from several points of view, starting with a classical imbedded version, and ending up with affine schemes.
• Properties of properties of morphisms: a note intended to help the reader organize common behaviors for properties of morphisms.
• 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 manifold.

### Problem sets

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 #2, due 1/23: In Chapter II of Hartshorne, do exercises 2.3 (b) and (c), 3.6, and 3.16.
Selected solutions for problem set #2.
• Problem set #3, due 1/30: In Chapter II of Hartshorne, do exercises 3.9, 3.10, and 3.13 (a)-(f).
Selected solutions for problem set #3.
• Problem set #4, due 2/6: In Chapter II of Hartshorne, do exercises 3.14, 4.2, and 4.3.
Selected solutions for problem set #4.
• 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.

### Final Paper

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 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 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.