## Math 248A - Algebraic Geometry Fall 2013

Instructor: Brian Osserman

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

CRN: 53790

Office: MSB 3218, e-mail:

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

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 will post lecture notes here on topics supplemental to what is covered by Hartshorne, most likely focused on abstract varieties and on providing a bridge from Chapter I to Chapter IV.

• 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.
• Projective varieties: A development of projective varieties as examples of abstract varieties, including a study of regular functions on projective varieties and morphisms to projective varieties. Also, a brief discussion of completeness.
• 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.
• 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.
• The Riemann-Roch and Riemann-Hurwitz theorems: We state the Riemann-Roch theorem without proof, and give a brief sketch of some applications, including a proof of the Riemann-Hurwitz theorem.

### Problem sets

Problem sets will be posted here on 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. Beyond this, you should not seek any external help.

• Problem set #1, due 10/7: Do Exercises 1.1 (part (c) is extra credit), 1.2, 1.3 and 1.4 of Chapter I of Hartshorne.
Selected solutions for problem set #1.
• Problem set #2, due 10/14: Do Exercises 1.5, 1.8, 1.10 and 1.12 of Chapter I.
Selected solutions for problem set #2.
• Problem set #3, due 10/21: Do Exercises 3.3, 3.11, 4.2 and 4.7 of Chapter I of Hartshorne, as well as the following exercises from lecture:
• morphisms from a quasiaffine variety X to affine n-space are in bijection with n-tuples of regular functions on X;
• if a morphism from X to Y has image contained in a subvariety Z, then the induced map X to Z is a morphism;
• the restriction map from the affine plane to the complement of the origin is an isomorphism on the rings of regular functions.
Selected solutions for problem set #3.
• Problem set #4, due 10/28: Do Exercises 3.15, 5.1, 5.2, 5.3, 5.4 (a) and (b), and 5.10 (b) and (c) of Chapter I of Hartshorne. For 3.15 (c), interpret "variety" to mean "quasiaffine variety".
Selected solutions for problem set #4.
• Problem set #5, due 11/4: Do Exercises 2.2, 2.5, 2.9 and 2.10 of the Abstract varieties via atlases lecture notes. Also:
• Prove or disprove and salvage if possible: a nonsingular affine algebraic set is irreducible.
• The only regular functions on the entire projective line (as defined in Example 1.7 of the lecture notes) are constant.
Selected solutions for problem set #5.
• Problem set #6, due 11/13: Do Exercise 3.1 of the Abstract varieties via atlases lecture notes, and Exercises 2.17 (a)-(b), 3.4 and 3.5 of Chapter I of Hartshorne. Also:
• Show that any prevariety X has a Noetherian topological space.
• If X is a prevariety and U an open subprevariety, then dim U = dim X. If Z is a closed subprevariety, then dim Z + codimX Z = dim X.
• Show that if X is a prevariety and Y is a variety, then two morphisms from X to Y which agree on a nonempty open subset of X must be the same.
Selected solutions for problem set #6.

• Problem set #7, due 11/18: Do Exercises 3.1 and 3.4 of the Projective varieties lecture notes.
Selected solutions for problem set #7.

• Problem set #8, due 11/25: Do Exercise 2.2 of the Nonsingular curves lecture notes, and Exercises 6.1, 6.2, 6.6(c) and 6.7 of Chapter I of Hartshorne.
Selected solutions for problem set #8.

• Problem set #9, due 12/2: Do Exercise 6.3 of Chapter I of Hartshorne, and also show that a morphism from the projective line to itself is ramified at all points if and only if it is ramified at infinitely many points, if and only if the characteristic is p and the morphism factors through the Frobenius (i.e., pth power) map.
Selected solutions for problem set #9.

### Final Exam

There will be a takehome final exam for the course.

The takehome final exam is due Wednesday, December 11 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.