November 12, 2016

GIT-quotient of affine variety

Let $G$ be a reductive group, $X$ a $G$-variety, $L$ a line $G$-bundle on $X$. Then the GIT-quotient is defined as
$$X //_L G:=\operatorname{Proj} \left( \bigoplus_{n \geqslant 0} H^0(X, L^{\otimes n})^G \right).$$
If $X$ is affine and $L=\mathcal O_X(\chi)$ for a character $\chi: G \to \mathbb G_m$, then the GIT-quotient may be considered as
  • first taking an affine quotient $\operatorname{Spec} (\mathbb k[X]^{G_\chi})$ by $G_\chi:=\operatorname{Ker}(\chi: G \to \mathbb G_m)$,
  • then taking a quotient by $\mathbb G_m=G / G_\chi$ via replacing $\operatorname{Spec}$ to $\operatorname{Proj}$ of non-negative gradings.
For the details on the construction see
Mukai An introduction to invariants and moduli, remark 6.14.

The first step is fairly stupid, but a proof that GIT-quotient is very general is given in
Mumford Geometric invariant theory, converse 1.13 of chapter 1.4.

From MO:253379.

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