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.
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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