$$X //_L G:=\operatorname{Proj} \left( \bigoplus_{n \geqslant 0} H^0(X, L^{\otimes n})^G \right).$$

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

The first step is fairly stupid, but a proof that GIT-quotient is very general is given in

Mumford

From MO:253379.

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.