An easy textbook for geometric invariant theory and moduli space which makes use of scheme theory 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.
 A: 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.
