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I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through the basics carefully and also give some idea of the link between deformation theory and intersection theory and/or K-theory. Texts with examples would be particularly nice. Applications to representation theory would also be useful to me.

As a specific example, I would like to understand how deformation theory helps in understanding the geometry of the Hilbert scheme of points on a surface (and the geometry of more general moduli spaces like Nakajima quiver varieties, if possible).

My background in geometry is Hartshorne and some material on constructible sheaves and D-modules. I've also come across some deformation theory from the representation theory side.

Thank you for the help.

Edit: I'm also perfectly happy with references that only deal with complex algebraic geometry.

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    $\begingroup$ Hartshorne has a book on Deformation Theory too. Have you looked at Greuel, Lossen, Shustin - Introduction to Singularities and Deformations (2007) ? $\endgroup$
    – M.G.
    Feb 5, 2016 at 21:49
  • $\begingroup$ I just took a look at it. It's not exactly what I'm looking for but it has some interesting stuff. The appendix looks useful. Thanks. $\endgroup$
    – Siddharth Venkatesh
    Feb 5, 2016 at 21:56
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    $\begingroup$ try Palamodov:mathnet.ru/php/… or some of the other references here: icmat.es/congresos/STM/abstracts/vanStraten.pdf and everyone bases their treatment on the 1964 Harvard PhD thesis of M. Schlessinger $\endgroup$
    – roy smith
    Feb 6, 2016 at 0:48
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    $\begingroup$ Perhaps you may try Eisenbud and Harris forthcoming book: isites.harvard.edu/fs/docs/icb.topic720403.files/book.pdf $\endgroup$
    – F Zaldivar
    Feb 6, 2016 at 3:24

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You could try Sernesi's book (deformations of algebraic schemes).

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