Instructor: Brian Osserman

Lectures: TuTh 2:40-4:00, MSB 2112.

CRN: 64051

Office: MSB 3218, e-mail:

Office Hours: Tu 4:10-5:00, Th 11:00-11:50

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 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.

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 varieties.

I shall, from time to time, post lecture notes on topics supplemental to what is covered by Hartshorne.

• Secant varieties and curves in projective space: this brief note uses secant varieties to construct morphisms from curves the projective spaces satisfying certain injectivity properties.
• 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 of Hartshorne.
• Problem set #5, due 2/11: Do Exercises 2.2 and 2.9 of Chapter II of Hartshorne.

• 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.

Guidelines:

• 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 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.