# 18.726 (Algebraic Geometry): Syllabus

Note: any changes or additions to this syllabus will be posted on the
main course web page: math.mit.edu/18.726.

### 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).
There may be a homework grader assigned based on class size and
availability, but I haven't been promised one yet.
### Lectures

Tuesdays and Thursdays, 11-12:30, room 4-261 (campus map). Note: MIT
protocol dictates that lectures start 5 minutes after the posted start
time and end 5 minutes before the posted end time.

### Textbooks

Required: Algebraic Geometry, by Robin Hartshorne.
(Warning: although this book is extremely useful, and you will learn a
lot from it, it is also a book that many love to hate, for
different and sometimes contradictory reasons!)
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.
(Most of those references will be to EGA1, which is also available in
English translation.) 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).

On top of all that, I expect to supplement
Hartshorne with some typed lecture notes. These will be distributed
in class and also via this web site.

### Prerequisites

18.725 or equivalent; since 18.725 includes 18.705 (commutative
algebra)
as a corequisite, that means 18.705 is an implicit prerequisite here.
More explicitly, a course in classical algebraic geometry at the level
of, say,
Shafarevich's Basic Algebraic Geometry I, or Part I of
Mumford's The Red Book of Varieties and Schemes. Some
familiarity
with (or at least lack of fear of)
homological algebra may come in handy later in the course.

If you did not take 18.725 (either last term or in a prior academic
year), please come talk to me at your earliest convenience, so I can
verify the "equivalence" of your background. One blanket warning:
18.725 covers
sheaves pretty thoroughly, so I will not be spending much time on them
at the beginning of the course; if you are not yet comfortable with
them,
you may start to have trouble in the course very soon!

### Homework

Problem sets will be assigned each Thursday to be turned in the
following Thursday; late homeworks will be accepted only by prior
arrangement (i.e., sometime before the due date). I tend to be pretty
reasonable about late homeworks; just don't push it. However, the
material in this course (especially the basic facility with
manipulating schemes that is crucial to further work in the subject) is
largely delivered through homework, so getting behind on the homework
is an extremely risky proposition!

The format for homeworks will be about the same as last term: on each
set you will be required to do about 10 problems out of a pool of
somewhat more. Unlike last term, some of the problems will not be given
explicitly, but only by reference into Hartshorne.

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
enrolled at MIT or a cross-registration school; in that case, see me
for guidance). If you do not intend to submit homeworks for grading,
you are still encouraged to register (or cross-register) as a listener.
If nothing else, that will make it more likely that the
class is assigned a grader.
### Exams

There are no exams in this course.

### Grading

The course grade will be based solely on the weekly problem sets; see
above.

### Syllabus and lecture calendar

The syllabus and (proposed) lecture calendar are here.