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I would like to study geometric invariant theory and moduli theory. It seems that a standard textbook for these fields is "Geometric Invariant Theory" written by D.Mumford, J.Fogarty and F.Kirwan. However, the book is difficult to read. Is there an easier alternative to this? I would like to read a textbook which makes use of scheme theory.

I read "Algbraic Geometry and Arithmetic Curves" written by Qing Liu from chapter 2.1 to chapter 4.2 and "Algebraic Geomtery" written by Hartshorne from chapter 3.1 to chapter3.4. Also, I understand Cartier divisor and Weil Divisor.

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    $\begingroup$ You can a look at books on invariant theory by Dolgachev and by Mukai. Although they might not fit all your requirements. $\endgroup$
    – Donu Arapura
    Commented Feb 3, 2023 at 8:46
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    $\begingroup$ Dolgachev's book is excellent and begins with a review of the classical approach to invariant theory (the "symbolic method"). There is a mistake in Mumford's book, and Dolgachev's book avoids that issue. Also, Dolgachev's book constructs the moduli spaces of Abelian varieties (and the moduli space of curves) in an "easier" way using the Kempf-Ness theorem (Dolgachev attributes the criterion to Kempf, but I believe it is Kempf-Ness). $\endgroup$
    – Jason Starr
    Commented Feb 3, 2023 at 11:05
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    $\begingroup$ What is a mistake in Mumford's book? $\endgroup$
    – YYY
    Commented Feb 3, 2023 at 12:17
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    $\begingroup$ Are you asking me where is the mistake in Mumford's book? $\endgroup$
    – Jason Starr
    Commented Feb 3, 2023 at 23:29
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    $\begingroup$ To Jason Starr. Yes, I am. If there is a mistake in Mumford's book, shouldn't I read his book? $\endgroup$
    – YYY
    Commented Feb 4, 2023 at 1:20

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The OP asked me about the mistake in Mumford's book (probably well-known to experts). I am attaching below something I wrote about this more than 10 years ago.

Dear Johan and Jarod,

At Stony Brook our student seminar has been going through GIT. While reviewing the proofs, I noticed something funny about Mumford's use of "uniform" as in "uniform categorical quotient". According to the definition, a morphism $f:X \to Y$ is a uniform categorical quotient if for every flat morphism $Y' \to Y$, forming the fiber product $X' = X \times_Y Y'$, the morphism $f': X' \to Y'$ is a categorical quotient. But then when Mumford proves certain morphisms $f$ are categorical quotients (e.g., the quotient of the semistable locus), he definitely needs that $Y$ is finitely presented (at least Noetherian), because he uses Noetherian induction, existence of closed points in each constructible set, etc. And then he asserts that this is a uniform categorical quotient, because you make the same argument after base change by $Y' \to Y$.

It seems to me the simplest "interpretation" is that Mumford actually only intends to allow flat morphisms $Y' \to Y$ which are composites of the following: (a) base changes obtained by extension of the ground field and (b) flat $k$-morphisms between finitely presented $k$-schemes. For all the applications we will have in the seminar, this is good enough. But I thought I would check with you in case (1) there is some better solution, or (2) this issue is well-known and discussed elsewhere.

Best regards,

Jason

Edit. Also there is a more substantial issue, below, spotted by Johan de Jong (the issue I spotted above is resolved by restricting to the category of finite type, separated schemes over a specified field).

http://www.math.columbia.edu/~dejong/wordpress/?p=76

Second edit. Going through my old e-mails from that student seminar, here are some other references.

Michel Brion, Introduction to actions of algebraic groups, Les cours du CIRM, 1 no. 1 (2010), 1-22.

Igor Dolgachev, Lectures on Invariant Theory, London Math. Soc. Lecture Note Series 296 (2003).

Peter Newstead, Geometric Invariant Theory, Lecture Notes CIMPA-2006.

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