## Math 248A - Algebraic Geometry Fall 2009

Instructor: Brian Osserman

Lectures: MWF 3:10-4:00pm, Physics-Geology 130

CRN: 43915

Office: MSB 3218, e-mail:

Office Hours: W 11:00-11:50, Th 3:10-4:00

Prerequisites: Math 250ABC, but please contact me if you are interested and have not taken these courses

Textbook: Hartshorne, Algebraic Geometry

Syllabus: We will cover the basics of classical algebraic geometry of affine and projective varieties defined by polynomial equations. See also the department syllabus.

Grading: 75% homework, 25% takehome final exam

Homework: Homework will be assigned roughly weekly

### Welcome to Math 248A: 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. Because of the classical focus for 248A, I hope it will be accessible to a broad audience, including applied math students.

### Lecture notes

I shall, from time to time, post lecture notes on topics supplemental to what is covered by Hartshorne. I see that a theme will soon emerge: Hartshorne omits a discussion of abstract varieties, and likewise omits a bridge from Chapter I to Chapter IV, so the notes will likely focus on filling in these topics.

• Recovering geometry from categories: this is a purely optional note for those who are categorically inclined, expanding on my remarks from class.
• Varieties via atlases: we describe how to define abstract varieties glued together from affine varieties, using atlases as in classical differential geometry.
• Nonsingular curves: we study abstract nonsingular curves, showing that they are always quasiprojective, and can always be compactified to nonsingular projective curves. Includes a brief section on normality and normalization.
• Chevalley's theorem and complete varieties: we first prove Chevalley's theorem on images of morphisms. We then define and explore the notion of completeness, analogous to compactness of topological spaces. We prove that completeness may be understood in terms of extending morphisms from nonsingular curves.
• Divisors on nonsingular curves: We introduce the concept of divisors on nonsingular curves, and study their properties and their relationship to morphisms to projective space.
• Differential forms: We introduce the concept of differential forms on nonsingular varieties, and then specializing to the case of curves, look at the associated divisors, as well as the relationship between differential forms and ramification of morphisms.

### Problem sets

Problem sets will be posted here on Fridays, due the following Friday in class. You are encouraged to collaborate with other students, as long as you do not simply copy their answers.

• Problem set #1, due 10/9: Do Exercises 1.1 (part (c) is extra credit), 1.2, 1.3, 1.4, 1.8, 1.10, and 1.12 of Chapter I.
• Problem set #2, due 10/16: Prove the following statements directly from definitions:
(a) if X and Y are affine varieties in A^n and A^m respectively, and f=(f_1,...,f_m) is a tuple of polynomials in n variables such that f(X) is contained in Y, then f gives a morphism from X to Y.
(b) If X is an affine variety, and f is a regular function on X, then f is a morphism to A^1.
Also do Exercises 3.1 (a) (replace A^1-{0} by Z(xy-1) in A^2), 3.2, 3.3, 3.11 (assume X is affine), 3.15, and 3.19 (part (b) is extra credit) of Chapter I.
• Problem set #3, due 10/23: Do Exercises 2.6, 2.7, 2.9, 2.16, 3.1 (b)-(d), 3.5 and 3.9 of Chapter I.
• Problem set #4, due 10/30: Do Exercise 3.14 of Chapter I, and Exercises 2.8, 2.10, 2.11 and 3.1 of the Varieties via atlases lecture notes.
• Problem set #5, due 11/6: Do Exercises 4.1, 4.2, 4.3, 4.7, 5.1 and 5.9 of Chapter I.
• Problem set #6, due 11/13: Do Exercises 3.2 and 3.3 of the Nonsingular curves lecture notes, and Exercises 6.1, 6.2 and 6.7 of Chapter I.
• Problem set #7, due 11/20: Do Exercises 4.4(c), 4.5, 5.10 and 6.6 of Chapter I.
• Problem set #8, due 12/2: Do Exercises 2.13 and 2.14 of the Chevalley's theorem and complete varieties lecture notes, and Exercise 1.8 of the Divisors on nonsingular curves lecture notes.

### Final Exam

The takehome final exam is due Wednesday, December 9 by 5:00. Obviously, you are not allowed to discuss it with any of your classmates or anyone else until after you have handed it in. You may email me with questions or concerns, but I will not give out any hints individually; if I determine that a hint is appropriate, I will email it to everyone.