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I am trying to read Mumford's Geometric Invariant Theory, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's Algebraic Geometry. Mumford cites a lot of results from EGA and SGA, but I cannot read French. Therefore, I want to ask:

What should I read if I want to read GIT?

I should mention that most of my knowledge is in differential geometry. I want to read Mumford's work to understand stable bundles and moment maps from another perspective; different from differential geometry.

Any suggestions for alternatives to Mumford's work will also be acceptable.

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  • $\begingroup$ Mumford has written a survey Stability of projective varieties in l'Enseignement Mathématique, Vol.23 (1977), which is much more accessible than his book. It is freely accessible on the web. $\endgroup$ – abx Apr 19 '19 at 14:59
  • $\begingroup$ Is there a specific part of GIT you’re having trouble with? $\endgroup$ – Samir Canning Apr 19 '19 at 15:10
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    $\begingroup$ @SamirCanning I just find too many unfamiliar terms and theorems. For example, Mumford used "Chevalley's Criterion" in page 6,(4). His reference is EGA ch. 4,14.4 which is written in French. I have never heard such a theorem. Also there are two many terms with prefix "Geometrically" such as geometrically injective, which means being injective on geometrcal points, but I am unfamiliar with such terms. I know I am very weak in algebraic geometry, so I want to know how to fill the gap. Anyway, thank you very much for your comment! $\endgroup$ – ZetaW Apr 19 '19 at 15:25
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Before I started reading/referencing Geometric Invariant Theory by Mumford, Fogarty, Kirwan I read Dolgachev's book Lectures on Invariant Theory. I also like Peter Newstead's book Introduction to Moduli Problems and Orbit Spaces. Given your stated interest, these treatments might be sufficient. Even if they are not, you should be able to read them after going through Hartshorne and then I would guess GIT will be "easier" to read (the technicality will still be the same, but your intuition will be better).

Addendum (given OP's comment): As far as a prerequisite for GIT itself, I don't know (beyond Hartshorne). I think when you come across a concept or term you are unfamiliar with you need to just "bite the bullet" and find a reference for that definition/concept and think about the concept. So you can use EGA/SGA as referenced directly in GIT or you can perhaps find what you need in the StacksProject (see this MO post about the pros/cons of this). But if you have first read, say Dolgachev's notes, then when you are trying to put together these new ideas into a coherent whole it will be easier since you will have some sense of what the bigger picture looks like.

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  • $\begingroup$ Thank you very much for your answer! I will try to read these books for a better intuituion. However I am also looking for a book which can serve a prerequisite of Mumford's book. Does there exist such a book? $\endgroup$ – ZetaW Apr 19 '19 at 16:01
  • $\begingroup$ @ZetaW I added an addendum that attempts to answer your comment. Others on MO probably can speak to this better than I. $\endgroup$ – Sean Lawton Apr 19 '19 at 16:17
  • $\begingroup$ Nolan R. Wallach has a new book "Geometric Invariant Theory Over the Real and Complex Numbers" that should be quite accessible. $\endgroup$ – M Mueger Apr 19 '19 at 18:08
  • $\begingroup$ I'm a little surprised that Kirwan's thesis "Topology of quotients in symplectic and algebraic geometry" hasn't been mentioned yet, as a place to see GIT in a rather differential-geometric perspective. Of course, Kirwan is coauthor on the 3rd edition of GIT. $\endgroup$ – Allen Knutson May 18 '19 at 23:05

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