18.727: Deformation theory and moduli spaces

This course was taught at MIT in fall 2000. Here are various notes in dvi, ps, and pdf formats.

Here's a detailed summary of the first lecture (dvi, ps, or pdf) (Sept. 7).
Here's a detailed summary of the second lecture (dvi, ps, or pdf) (Sept. 12).
I hope to put up notes for classes 3 and 4 soon (in the next couple of weeks).
Here's a detailed summary of the fifth lecture (dvi, ps, or pdf) (Sept. 21).
Lecture 6 (Sept. 26) notes weren't TeXed and are long lost. (Similarly for later notes.)
Here are some notes from the seventh lecture (dvi, ps, or pdf) (Sept. 28).
Lecture 8 (Oct. 3).
Here are some notes from the ninth lecture (dvi, ps, or pdf) (Oct. 5).
Here are some notes from the tenth lecture (dvi, ps, or pdf) (Oct. 12).
Here are some notes from the eleventh lecture (dvi, ps, or pdf) (Oct. 17).
Here are some notes from the twelfth lecture (dvi, ps, or pdf) (Oct. 19).
I don't have TeXed notes from the thirteenth lecture. Eventually I might type some up and post them here.
Here are some notes from the fourteenth lecture (dvi, ps, or pdf) (Oct. 26).
Here are some notes from the fifteenth lecture (dvi, ps, or pdf) (Oct. 31).
Here are some notes from the sixteenth lecture (dvi, ps, or pdf) (Nov. 2).
Here is a handout with Schlessinger's theorem, to save me from having to keep writing it on the board (dvi, ps, or pdf).
Here is another example of why you want to think about moduli groupoids (i.e. to each base, you think of families along with their automorphism groups) rather than moduli functors (families up to isomorphism), from Jim Borger. It fits best with the earlier part of the course, but came up here, because Schlessinger just deals with the latter.
The seventeenth lecture (Nov. 7) will be given by Jason Starr.
Here are some notes from the eighteenth lecture (dvi, ps, or pdf) (Nov. 9).
Here are some notes from the nineteenth lecture (dvi, ps, or pdf) (Nov. 14).
The twentieth lecture began with a guest lecture by Ragni Piene, visiting from Norway. Here are some notes from my portion. (dvi, ps, or pdf) (Nov. 16).
Lecture 21 (Nov. 21).
Lecture 22 (Nov. 28).
Roy Skjelnes will be the guest lecturer for the twenty-third lecture (Thurs. Nov. 30). A sketch of what he might talk about:
To describe the Hilbert scheme of Spec( k[x]/(x^n)), a think point on the line, and the Hilbert scheme of Spec( k[x]_{(x)}), the local ring of a point on the line. One will then get concrete examples of Hilbert schemes and how they satisfy their universal properties, and examples of how little the set of k-rational points may tell about the Hilbert scheme.
His notes are here: (dvi, ps, or pdf).
Lecture 24 (Dec. 5).
Lecture 25 (Dec. 7), the last lecture.

Suggested references:

There is no perfect universal reference. The best references are those with a narrow specific goal. Here are a few; these are off the top of my head, so I've probably forgotten something important. I'll add more when I have time.

For stacks, Laumon and M-B is the canonical (hard) reference; the upcoming book of Behrend-Edidin-Fantechi-Gottsche-Fulton-Kresch will be great, but is still a couple of years away. Deligne and Mumford's original paper (on the irreducibility of the moduli space of curves) is still quite readable (but has gaps). The appendix to Vistoli's Inventiones article of '89 is a great detailed summary. Another excellent exposition is Edidin's notes on math.AG. Mumford's article "Picard groups of moduli problems" is also very explicit and beautiful.

I like Vistoli's notes "The deformation theory of complete intersections" very much (see math.AG). Schlessinger's original Tr.A.M.S. article and his Ph.D. thesis are both foundational. Sernesi has some notes that are excellent too; I hear they may be turned into a book soon.

In "Rational Curves on Algebraic Varieties", by the Mighty Kollár, has a great section on deformation theory and Hilbert schemes. "Curves on Algebraic Surfaces" by the Amazing Mumford is also fundamental .

Buchweitz and Flenner have a book coming out on deformation theory that promises to be very good.

Mike Artin has two good sources: the book "Lectures on deformations of singularities", and the '74 Inventiones article "Versal deformations and algebraic stacks".

Illusie's two volumes are the canonical deformation theory source, but they are very abstract.

In the analytic category, see Kodaira.

Deformation theory exercise: who is this?:

Hint: his name appears earlier on this page. (Thanks to Bobby Kleinberg and Tara Holm for this image.)

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