Spring 2017-2018

**Instructor:** Dan Abramovich

**Class meeting:** Mondays, Wednesdays and Fridays at 11:00-11:50am in Kassar 105

**Office:** Kassar 112 / 118

**Telephone:** (401) 863 7968

**E-mail:** `abrmovic ( at ) math (period) brown (dot) edu`

**Web site:**
`http://www.math.brown.edu/~abrmovic/MA/s1718/index.html`

**VERY Preliminary Office hours:** Monday, Wednesday 10-11 am; Friday 2:00-3:00, or
by appointment (I sometime require ofice hour attendance)

**Homework page:** here

**Presentation page:** here

**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 Kollar

**Subjects**

I will start with as much of sheaves of modules
and differentials as needed for the rest. Then we'll go through
resolution of singularities in characteristic 0 following Kollár. Then
we'll do divisors and algebraic curves and proceed as time permits.

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traditionally considered scary.** This is an outdated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 my ICM talk.

**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 January 24.

Add date is February 6. Grade option date is
February 21 and Audit date March 9.

Between February 12 and 16 there will be
special lectures and no office hours.

February 19 Brown is out for
President's Day.

Spring break March 26-30.

let me know if you want to go to
AGNES April
13. (Registration deadline has passed.)

Watch for possible updates regarding April 16-20.

Last class May 7 (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