Math 248B - Algebraic Geometry Winter 2014

Instructor: Brian Osserman

Lectures: MWF 3:10-4:00, Physics 140.

CRN: 84071

Office: MSB 3218, e-mail:

Office Hours: M 1:10-2:00, F 2:10-3:00

Prerequisites: Math 248A or equivalent

Textbook: Hartshorne, Algebraic Geometry

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.

• Recovering classical notions of points and morphisms in the scheme setting: discusses how morphisms of prevarieties and classical notions of points of affine varieties over non-algebraically fields can be understood in the scheme setting, with the help of the concept of a scheme over a base.
• 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.

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 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.