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:**
`http://www.math.brown.edu/~abrmovic/MA/s1920/index.html`

**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

**Notes**

Sheaves of modules

Introduction to resolution of singularities

A few presentations following Kollár

Draft presentation 3.10 following Kollár

**Subjects**

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traditionally considered scary.** This is an outdated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.

**Collaboration**

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

http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html