The invariant section ring and the projective GIT quotient

Assume the Axiom of Choice inherited from the Proj construction. Let $G$ be a complex affine algebraic group acting algebraically on a complex projective variety $X$, let $L$ be an ample $G$-linearized invertible sheaf (G-linearizations of invertible sheaves on a complex G-variety, Absolute ampleness by affine section opens), and let $R(X,L)={⨁}_{n\ge 0}\mathrm{\Gamma }(X,L^{\otimes n})$ be its graded section ring (Linearizations of tensor powers and the equivariant section ring). Write $R(X,L{)}^G={⨁}_{n\ge 0}\mathrm{\Gamma }(X,L^{\otimes n}{)}^G$ for the graded subalgebra of $G$-invariant sections.

The projective GIT quotient of $X$ by $G$ with respect to $L$ is the $\mathbb{C}$-scheme $X\mathrm{/}\text{\hspace{0.17em}⁣}{\mathrm{/}}_{L}G:=\mathrm{Proj}R(X,L{)}^G$ (Points of Proj of a graded ring, Proj carries a scheme structure), the Proj of the graded invariant subalgebra.

For a homogeneous invariant section $f\in \mathrm{\Gamma }(X,L^{\otimes d}{)}^G$ with $d\ge 1$ write $D_{+}(f)=\mathrm{Spec}(R(X,L{)}^G{)}_{(f)}\subseteq X\mathrm{/}\text{\hspace{0.17em}⁣}{\mathrm{/}}_{L}G$ for the standard open chart of $\mathrm{Proj}$, and $X_f=\{x\in X:f(x)\mathrm{\ne }0\}$ for the nonvanishing locus of $f$.

The construction is recorded together with the given linearization and the ample sheaf $L$; replacing $L$ by a positive tensor power does not change the Proj (Proj is invariant under Veronese regrading).

• Ampleness. The definition uses the ampleness of $L$ only to know that the section ring has sufficiently many sections for the construction to be the GIT quotient; the Proj itself is defined for any graded algebra, and, when $G$ is reductive, the quotient properties of $X\mathrm{/}\text{\hspace{0.17em}⁣}{\mathrm{/}}_{L}G$ are proved in Projective GIT quotient for a linear action and The projective GIT quotient from the invariant section ring.
• Finite generation is not asserted here. The definition does not claim that $R(X,L{)}^G$ is finitely generated, nor that $X\mathrm{/}\text{\hspace{0.17em}⁣}{\mathrm{/}}_{L}G$ is of finite type; For reductive $G$, both are proved in the two theorems just named, the first in the linear case and the second in the ample case.
• Dependence on the linearization. The quotient genuinely depends on the chosen linearization of $L$ and not only on the isomorphism class of $L$; this is recorded in G-linearizations of invertible sheaves on a complex G-variety and demonstrated on the companion page by The semistable locus depends on the linearization, not only on the sheaf ↗ and GIT quotients of the projective line for different linearizations ↗.
• Veronese invariance. Replacing $L$ by $L^{\otimes m}$ replaces $R(X,L{)}^G$ by its $m$-th Veronese subalgebra and leaves $\mathrm{Proj}$ unchanged (Proj is invariant under Veronese regrading).