Math 6170 (Fall 2023):
Algebraic Geometry I

About this course

Welcome to Algebraic Geometry I (Math 6170)! This is a first-semester graduate algebraic geometry course. We will begin with an introduction to the subject through the study of algebraic plane curves. Next we will study varieties in affine space and in projective space, using sheaves but not (yet) using schemes. We will conclude with an introduction to schemes.

Contact information

• Instructor: Jonathan Wise
• Office: Math 204
• E-mail: jonathan.wise@colorado.edu
• Phone: 303 492 3018

You may also contact me anonymously.

Office hours

My office is Room 204 in the Math Department. My office hours sometimes change, so I maintain a calendar showing the times I will be available. You can also make an appointment or drop in without an appointment.

Syllabus and textbook

We will follow Andreas Gathmann's notes on algebraic curves and on algebraic geometry. Here is an ambitious syllabus (we will probably fall behind):

• Week 1 (18 January — 20 January): affine curves and intersection multiplcity (curves, §§0 — 2)
• Week 2 (23 January — 27 January): projective curves and Bézout's theorem (curves, §§3 — 5)
• Week 3 (30 January — 3 February): divisors and Riemann–Roch (curves, §§6 — 8)
• Week 4 (6 February — 10 February): affine varieties and the Zariski topology (ag, §§0 — 2)
• Week 5 (13 February — 17 February): the structure sheaf and morphisms (ag, §§3 — 4)
• Week 6 (20 February — 24 February): varieties and projective space (ag, §§5 — 6)
• Week 7 (27 February — 3 March): projective space and Grassmannians (ag, §§7 — 8)
• Week 8 (6 March — 10 March): blowing up and smoothness (ag, §§9 — 10)
• Week 9 (13 March — 17 March): cubic surfaces and other examples (ag, §11)
• Week 10 (20 March — 24 March): schemes (ag, §12)
• Week 11 (3 April — 7 April): sheaves of modules (ag, §13 — 14)
• Week 12 (10 April — 14 April): differentials and smoothness (ag, §15)
• Week 13 (17 April — 21 April): cohomology of quasicoherent sheaves (ag, §16)
• Weeks 14 and 15 (24 April — 3 May): TBD

The course has approximately 3 parts. In the first 3 weeks, we are going to try to get familiar with the questions and techniques of algebraic geometry in the most familiar possible context: plane curves. In the second part, we will learn about the main objects of algebraic geometry, algebraic varieties, and some of the ways they are constructed. Finally, we will learn about schemes, which are the modern language of algebraic geometry. If we actually manage to stick to the above schedule, possible topics for the last two weeks include Serre duality and revisiting (and generalizing) our study of curves.

Prerequisites

You will need some comfort with commutative algebra. Familiarity with complex analysis will help a lot with intuition.

Course goals

The goal is to develop enough comfort with the language of algebraic geometry to be able to read a book like Hartshorne's, Vakil's, or Griffiths and Harris's.

Homework and grading

There will be daily reading assignments and weekly homework assignments (that you may submit in groups). You will also be expected to participate in the online discussion on Discord. For the weekly homework assignments, you may do as many or as few problems from the week's material as you like, but you should do some every week.

For graduate students: If you complete 2/3 or more of the assignments (including homework and Discord discussion), you will receive an A. If you complete between 1/3 and 2/3 of these assignments, you will receive an A-. As long as you make a reasonable effort in the course, you will receive at least a B+.

For undergraduate students: If you complete 2/3 or more of the assignments (including homework and Discord discussion), you will receive an A. If you complete between 1/3 and 2/3 of these assignments, you will receive a B. As long as you make a reasonable effort in the course, you will receive at least a C.

Feedback

Special accommodations, CoViD-19, classroom behavior, and the honor code

The Office of Academic Affairs officially recommends a number of statements for course syllabi, all of which are supported in this class.

If you need special acommodation of any kind in this class, or are uncomfortable in the class for any reason, please contact me and I will do my best to remedy the situation. You may contact me in person, by e-mail, or anonymously.