18.721 - Introduction to Algebraic Geometry


SYLLABUS

Course Overview

When you have completed this course, you will be well prepared for a graduate course in algebraic geometry, and you should be able to read some papers in the subject. For this, it is essential that you become familiar with cohomology. The effort required is worthwhile. Though algebraic geometry is usually taught assuming familiarity with commutative algebra, we won’t assume things beyond 18.702 (Algebra II) are known, and we will keep commutative algebra at a minimum. To help make the material accessible, we’ve made some simplifying restrictions: The most important are:

  1. We work exclusively with quasiprojective varieties over the field of complex numbers.
  2. Theorems may not be stated or proved in their most general form.
  3. Cohomology is introduced only for modules (aka quasicoherent sheaves).

The course begins with plane curves. They provide a good introduction, and as we will see, they govern the geometry of varieties of any dimension. It would be reasonable to spend a whole semester on plane curves, but we won’t do that.

Texts

We will follow the notes written for the course:

  1. Notes for the course by Michael Artin (errata)

Please let us know if you find any typos.

In addition to the notes, we also recommend the following references:

  1. Algebraic Curves by William Fulton (pdf)
  2. Plane Algebraic Curves by Gerd Fischer (AMS bookstore)
  3. Algebraic Varieties by George Kempf

Fulton is the book closest to the content of the course that is available. It can be downloaded free of charge at the link above.

Problem Sets

Weekly problem sets will be posted online. Solutions are to be prepared in typeset form (typically via LaTeX) and submitted electronically as a pdf file via Gradescope by midnight on the due date (the first problem set will be due Friday, September 17). Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty.

Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing Sources consulted: none at the top of your submission.

Final project

A final research project will be assigned in place of a final or exams. In consultation with the instructors, students will choose a chapter or research article related to the course but outside the syllabus. They will write and present an article summarizing (an interesting part of) its contents, pose a research question naturally arising in this work, and then, if time permits, try to answer it. The approximate length should be 5-20 pages, and the intended audience is your peers. Depending on your choice of topic, you may get quite far into this sequence, or the summary itself may comprise the majority of your paper.

Project plan/outline due by November 15th. Possible topics

Grading

Grades will be based on your performance on the problem sets (77%) and the final project (23%). The lowest score of one of your problem sets score on be dropped. (This includes any problem set you did not submit.) There are no exams and no final exam.

Disability Accomodations

Please contact Student Disability Services as early in the term as possible, if you have not already done so. If you already have an accommodation letter, please be sure to submit a copy to Mathematics Academic Services in room 2-110. Even if you do not plan to use any accommodations, if there is anything we can do to facilitate your learning, please let us know.