18.726 (Algebraic Geometry): Syllabus
This syllabus is subject to change until the semester begins.
After the semester begins, changes to this syllabus will be posted on the
course web page. There is a separate
list of topics.
About the class
It is easiest to explain what 18.726 is about by contrasting it with
18.725.
In 18.725, one studies algebraic geometry using powerful techniques but
with a "classical" frame of reference set up in the early 20th century.
In the 1960s, a school of mostly French mathematicians led by
Grothendieck
developed a new language and toolkit for dealing with algebro-geometric
objects, centered around the notion of a scheme. The
construction
of a scheme is the algebraic geometry analogue of the construction of
a manifold (or differentiable manifold, or...) by glueing together
small
easily understood pieces; working with schemes makes it possible to
distinguish "local" and "global" phenomena as is typical in other
types of modern geometry. The definition of an "abstract algebraic
variety" from 18.725 approximates this, but the notion of a scheme
is much more general and much more flexible; it correctly accounts for
nilpotents (a big help to intersection theory), deals well with base
fields which are not algebraically closed, and easily accommodates
objects of number-theoretic interest (like the ring of rational
integers).
In some ways, this course is a language class: we will be learning how
to read, write and speak the language of schemes. Along the way, we
will absorb some of Grothendieck's insights into this language: the
importance of working locally, the relevance of relative properties of
morphisms, base change, and how to think algebraically about the
cohomology of algebro-geometric objects.
Instructor
Kiran Kedlaya, 2-165, x3-2946, kedlaya[at]mit[dot]edu (web site, more contact info).
Office hours Thursday, 2:30-3:30.
There may be a homework grader assigned based on class size and
availability.
Lectures
Mondays, Wednesdays, and Fridays, 10-11, room 2-102.
I will follow the MIT
convention of starting my lectures 5 minutes after the posted start time,
and ending them 5 minutes before the posted end time.
Textbooks
Required: Algebraic Geometry, by Robin Hartshorne.
This is required mostly because I will draw many of the homework problems
from it; I plan to follow it less closely than the last time I taught
this course. Instead, I will distribute typed lecture notes;
a good model for what I plan to do is given by
this past course
by Ravi Vakil.
Recommended: download Éléments de
Géométrie Algébrique,
by Grothendieck and Dieudonné,
from NUMDAM, and keep it on hand
in case you need to look up one of the odd references to it I will be
sprinkling throughout the course. (See the main web page for precise
links.) However, do not attempt to read EGA
directly; this has
been likened to reading a dictionary, with similarly unfruitful
results.
On the other end of the intuitive-to-formal spectrum,
you may also find helpful The Geometry of Schemes, by
Eisenbud and Harris; it provides some intuition lacking in Hartshorne
(e.g., the functor of points).
Other possible references include
Mumford's The Red Book of Varieties and Schemes
and Ueno's Algebraic Geometry (all three volumes).
On top of all that, I am planning to distributed typed lecture notes.
Prerequisites
Required: 18.705 (commutative algebra) or equivalent;
18.725 (the first semester algebraic geometry course) or equivalent.
A suitable equivalent for 18.725 is familiarity with
Shafarevich's Basic Algebraic Geometry I, or Part I of
Mumford's The Red Book of Varieties and Schemes.
Recommended: some exposure to homological algebra (e.g., in 18.905)
and rudimentary category theory (as in 18.725 last semester, or 18.905).
I'll introduce what I need as I go along, but possibly in a pretty sketchy
way with a lot of assertions left as "exercises for the listener".
Homework
Problem sets will be assigned each Friday, to be turned in the following
Friday. Each problem set will involve doing a certain number of problems
chosen from a larger set of offerings; students should choose what to
submit based on their own background and interests.
You should plan to spend a lot of time on the homework,
since much of the challenge of this material is getting comfortable
with the basic notions, and this is best achieved by grappling with them
at close range.
If you need to turn in a problem set late, please arrange an extension
with me beforehand. If you get too far behind, I may ask you to drop
the course.
Note that if you intend to submit homeworks for grading, you must be
enrolled in the class unless that is impossible (e.g., if you are not
a student, or are enrolled at a school that does not permit
cross-registration at MIT).
This is needed to ensure accurate enrollment figures so that we
can be assigned a course grader.
If you do not intend to submit homeworks for grading,
you may register (or cross-register) as a listener.
Exams
There are no exams in this course.
Grading
The course grade will be based solely on the weekly problem sets; see
above.
Note for undergraduates
This class is not particularly intended for undergraduates, and is not
appropriate as a first course in algebraic geometry. (Remember that 18.725
and 18.705 are both prerequisites.) Also, the time required to complete
the homework in this class may seem large even compared to other graduate
courses. However, undergraduates
with adequate preparation will be permitted to enroll; I will be the sole
judge of what constitutes adequate preparation. Grading criteria will not
distinguish between undergraduate and graduate students (see above).
If you are an
undergraduate (at MIT or elsewhere) wishing to take the course for a grade,
please send me an email with a detailed description of the background
which you think makes you prepared to take the course.
I may ask you to meet me in person to evaluate further.