MA 206 0 Algebraic Geometry
Spring 2019-2020

Instructor: Dan Abramovich
Class meeting: Mondays, Wednesdays and Fridays at 10:00-10:50am in Kassar 105
Office: Kassar 118
Telephone: (401) 863 7968
E-mail: abrmovic ( at ) math (period) brown (dot) edu
Web site:
VERY Preliminary Office hours: Monday, Wednesday 11-11:50 am; Friday 2:00-3:00, or by appointment (I sometime require ofice hour attendance)
Homework page: here
Presentation page: here
Canvas: direct link.

Text: Algebraic Geometry and arithmetic curves by Qing Liu, Oxford Graduate Texts in Mathematics, ISBN 978-0199202492.
With a Brown University login you can read the current edition of the book online. Individual chapters of the previous 2002 edition may be downloaded in PDF.
The author's list of errata here.

Supplementary textbooks

Ravi Vakil, The rising sea: Foundations of algebraic geoemtry.

János Kollár, Lectures on resolution of singularities, Princeton University Press, also available through Brown's library web page using your Brown login.

Algebraic Geometry by Robin Hartshorne, Springer-Verlag, Cor. 8th printing, 1997, ISBN: 0-540-90244-9

Sheaves of modules

Introduction to resolution of singularities

A few presentations following Kollár

Draft presentation 3.10 following Kollár

Crash course on cohomology

I will start with as much of smoothness and modules of differentials as needed for the rest. Then we'll go through resolution of singularities in characteristic 0 following Kollár, with supplements on recent progress. Then we'll do sheaf cohomology. Then if time permits we'll do divisors and algebraic curves.

The subject of resolution of singularities in characteristic 0 was traditionally considered scary. This is an outdated view which no longer is based on reality. One of my goals is that you will come to know that resolution of singularities - in characteristic 0 - is not so hard. When you go out into the world people will think you have superpowers.

Another goal is for you to help me prepare for an Obewolfach Seminar in fall 2020.

Basis for grading
When we study material in Liu's book I'll assign weekly homework. When we study resolution of singularities, algebraic curves, and other topics if time permits, students will also present material from the sources in class and submit their notes. Grades will be based on these solutions, presentations and notes.

Solutions to problem sets are to be stapled.


You are encouraged to collaborate with other students in the class on your homework, although I suggest that you think carefully about each problem on your own first. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)
You are also encouraged to collaborate on preparation of presentations.

How much time will this class take?

Roughly speaking, you should expect to spend twelve hours every week outside of class, including attending office hours, reviewing class material, doing problem sets, preparing presentations. In addition to three hours of class every week, I estimate a total of 15*13 = 195 hours of time spent on this class.

Dates (see also the academic calendar)

Class starts Wednesday January 22.
Add date is February 4. Grade option date is February 19 and Audit date March 6.
Monday February 17 (and the following day) Brown is out for President's Day.
Spring break March 23-27.
let me know if you want to go to
AGNES March 27. (Registration deadline February 15.)
Last class May 4 (during reading period).

Accommodations for students with disabilities

Please contact me as early in the semester as possible so that we may arrange reasonable accommodations for a disability. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 401-863-9588 or online at