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.


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.


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.


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.


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".


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.


There are no exams in this course.


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.