Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces.

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When are GIT quotients projective?

Some background on GIT Suppose G is a reductive group acting on a scheme X. We often want to understand the quotient X/G. For example, X might be some parameter space (like the space of possible ...
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Why can we define the moment map in this way (i.e. why is this form exact)?

Given a symplectic manifold $(X, \omega)$ and a group $G$ acting on $X$ preserving the symplectic form, we define the moment map $\mu : X \to \mathfrak{g}^*$ so that $$ \langle d\mu(v), \xi\rangle = ...
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Can a coequalizer of schemes fail to be surjective?

Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the coequalizer of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ ...
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Toric varieties as quotients of affine space

One way to define toric varieties is as quotients of affine $n-$space by the action of some torus. However, this is not strictly true as we need to throw away "bad points" which ruin this ...
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Understanding the definition of the quotient stack $[X/G]$

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles. Explicitly, let $G$ be an affine smooth group ...
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A line bundle that does not admit a G-linearisation

I have been thinking about quotients lately and pondered the following: Let $G$ be a connected linear algebraic group and $X$ a $G$-variety acting via the morphism $\sigma:G\times X\rightarrow X$. ...
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Are orbits of an affine algebraic monoid affine?

Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...
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Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...