Introduction to Algebraic Geometry

Spring 2008

Course number: 22M:330:002 Topics in Algebra

Class meets: TuTh 1:05-2:20pm in 221 MacLean Hall.

Instructor: Julianna Tymoczko
Office: 225G MacLean Hall
Office Phone: 335-0790
email: tymoczko AT math DOT uiowa DOT edu.

Office hours: Monday 10-11:30am, Tuesday 2:30-4pm, and by appointment.

Course homepage: http://www.divms.uiowa.edu/~tymoczko/ag_intro/index.html This webpage will be updated regularly with suggested problems and readings.

Prerequisites: Abstract algebra at the first-year graduate level. Algebraic topology at the first-year graduate level is helpful but not necessary.

Course description: Algebraic geometry has two very different faces. Classical algebraic geometry reached a pinnacle in the Italian school of the late nineteenth century, though it is now experiencing a renaissance. It attacks concrete questions, for instance how many lines there are on a cubic surface, in hands-on ways involving a lot of elbow grease. (By the way, the answer is 27.)

By the turn of the century, these methods had carried algebraic geometry about as far as they could. The twentieth century saw the birth of abstract algebra (groups, rings, modules), which was the key to a new algebraic geometry. In the 1960s, Grothendieck (with others) discovered modern algebraic geometry in schemes. The notion of schemes unifies algebraic geometry and number theory, and is an incredibly powerful tool in contemporary mathematics. For instance, Andrew Wiles's proof of Fermat's Last Theorem used these tools in crucial ways, and modern cryptography is based on algebro-geometric techniques.

This course will present both aspects of algebraic geometry. The first ten weeks will emphasize classical algebraic geometry, including many examples, to develop intuition and give a feel for the field. The last third of the course will discuss schemes, to introduce this transformational concept of modern mathematics.

Goals and objectives: The goal of the course is to give an introduction to the main topics and ideas in contemporary algebraic geometry, including the important questions that are asked in algebraic geometry and the kinds of methods that can be used to solve them. At the end of the semester, a student who bumps into these concepts in his or her research would know what they mean and how to get more information if needed. Students will also gain some practice with the basic tools and techniques of algebraic geometry.

Course work: Regular problem sets will be handed out, on the grounds that math is learned through doing not listening. Each set will be short and will consist of problems geared to consolidate your understanding of the material. No homework will be collected.

Each student will give an in-class presentation about a particular algebraic variety or family of algebraic varieties, selected from the student's research interests or from a list provided by the instructor. Possibilities include Veronese and Segre embeddings, Grassmannians, flag varieties, curves, toric varieties, and Hilbert schemes of points in the plane.

In addition, the students will collectively produce a LaTeXed set of course notes that will be posted on the course webpage. Each student will be responsible for LaTeXing at least one lecture to contribute to the notes. This page has some useful LaTeX links and encouragement.

The presentation, together with class participation, will determine the course grade.

Readings (suggested but not required):

[D] Danilov, Algebraic Varieties and Schemes, in Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes (ed. Shafarevich)

[EH] Eisenbud and Harris, The Geometry of Schemes

[GH] Griffiths and Harris, Principles of Algebraic Geometry

[Harr] Harris, Algebraic Geometry: A First Course

[Hart] Hartshorne, Algebraic Geometry

[R] Reid, Undergraduate Algebraic Geometry

[S] Shafarevich, Algebraic Geometry I: Varieties in Projective Space

All readings are on reserve in the Mathematical Sciences Library.

A note to students thinking ahead: Reid's book is a particularly readable introduction to algebraic geometry if you are interested in getting a jumpstart on the course.