The projective GIT quotient from the invariant section ring

Finite generation and projectivity. By F1 the invariant section ring $R^G$ is a finitely generated graded $\mathbb{C}$-algebra with degree-zero part $\mathbb{C}$, so $Y=\mathrm{Proj}{R}^G$ is projective of finite type. This proves (i).

Choose an embedded model. Fix any equivariant closed immersion $i:X\hookrightarrow \mathbf{P}(V)$ with $i^{\ast }\mathcal{O}(1)\cong L^{\otimes m}$ as $G$-linearized invertible sheaves, whose existence is F2. Let $X^{\mathrm{\prime }}=i(X)$, $S=\mathbb{C}[V]=\mathrm{Sym}(V^{\ast })$, and let $A\subseteq R(X,L^{\otimes m})$ be the homogeneous coordinate ring, the image of the restriction map $S\to R(X,L^{\otimes m})$.

The full section ring is finite over $A$. For $0\le j<m$, the graded module $M_j={⨁}_{k\ge 0}\mathrm{\Gamma }(X,L^{\otimes (mk+j)})$ is the section module of the coherent sheaf $i_{\ast }{L}^{\otimes j}$ on $\mathbf{P}(V)$, so it is finite over $S$ by F5. If a homogeneous polynomial in $S$ restricts to zero on $X$, then it acts by zero on each $M_j$, since multiplication by it is restriction followed by multiplication of sections. Thus the $S$-actions on the $M_j$ factor through $A$, each $M_j$ is a finite $A$-module, and $R={⨁}_{j=0}^{m-1}{M}_j$ is a finite $A$-module. By the faithful-module criterion in F5 every homogeneous $\sigma \in R$ is integral over $A$.

Invariant sections give invariant coordinate charts. Let $\sigma \in R_n^G$ be homogeneous of positive degree. Since it is integral over $A$, it satisfies a monic relation over $A$; taking the homogeneous component of total degree $rn$ gives a relation ${\sigma }^r+{\sum }_{i=1}^r{a}_i{\sigma }^{r-i}=0$ with $a_i\in A\cap R_{in}$ (zero when $m\nmid in$). Apply the Reynolds operator of $R$ to this relation. Its naturality under multiplication by the invariant $\sigma$ and under the inclusion $A\hookrightarrow R$ gives ${\sigma }^r+{\sum }_{i=1}^r{R}_A(a_i){\sigma }^{r-i}=0,$ where $R_A(a_i)\in A^G\cap R_{in}$ by F7. If $\sigma (x)\mathrm{\ne }0$ at $x\in X$, not all $R_A(a_i)$ can vanish at $x$, since evaluating the displayed relation in the one-dimensional fiber of $L^{\otimes rn}$ would otherwise give $\sigma (x{)}^r=0$. Hence every point semistable for the full section ring lies in a nonvanishing chart of a positive-degree invariant in $A^G$. The reverse inclusion is immediate from $A^G\subseteq R^G$, so the embedded semistable locus $X^{\mathrm{\prime }ss}$ equals $X^{ss}(L)$. The same monic relation shows that the charts $D_{+}(f)$ with $f\in A_{+}^G$ cover $\mathrm{Proj}{R}^G$: for any homogeneous prime avoiding $R_{+}^G$, choose $\sigma \in R_{+}^G$ outside it; some coefficient $R_A(a_i)$ in its relation must also be outside the prime.

Identify the two quotient targets chartwise. For $f\in A_{+}^G$, the affine chart $X_f$ has ring $A_{(f)}$ by F4. Fractions in $R_{(f)}$ are regular on $X_f$, and F6 shows that every regular function there is such a fraction, so $A_{(f)}=\mathrm{\Gamma }(X_f,{\mathcal{O}}_X)=R_{(f)}$ as $G$-algebras. Taking invariants and using F7 gives $(A^G{)}_{(f)}=(R^G{)}_{(f)}$. The $D_{+}(f)$ for $f\in A_{+}^G$ cover both $\mathrm{Proj}{A}^G$ and $Y=\mathrm{Proj}{R}^G$ by step 2.1; these identical chart rings and their localization maps therefore glue to a canonical isomorphism $\theta :\mathrm{Proj}{A}^G\stackrel{\sim }{\to }Y$. On every chart the quotient morphisms from $X_f$ are induced by the same inclusion of invariant regular functions into $\mathrm{\Gamma }(X_f,{\mathcal{O}}_X)$, so $\theta$ identifies the linear-action quotient with the section-ring quotient.

Compare positive tensor powers and embedded data. For $r\ge 1$, the $r$-th Veronese of $R^G$ is $R(X,L^{\otimes r}{)}^G$, so F9 identifies the Proj quotient data for $L$ and $L^{\otimes r}$. The nonvanishing locus of a section equals that of every positive tensor power, so the semistable loci coincide; closedness of orbits in that same locus and finiteness of stabilizers then give equality of the stable loci. For the fixed compatible embedding in step 1.2, step 2.1 proves that the definitions on $X^{\mathrm{\prime }}=i(X)$ agree with those from the full section ring. Moreover the equivariant surjection $S\to A$ induces a surjection $S^G\to A^G$ by F7; hence invariant forms on $X^{\mathrm{\prime }}$ lift to invariant forms on $\mathbf{P}(V)$, giving $X^{\mathrm{\prime }ss}=X^{\mathrm{\prime }}\cap \mathbf{P}(V{)}^{ss}$. The orbit closure of a point of $X^{\mathrm{\prime }}$ in the ambient semistable locus stays in $X^{\mathrm{\prime }}$, because $X^{\mathrm{\prime }}$ is closed and invariant; closedness there is therefore equivalent to closedness in $X^{\mathrm{\prime }ss}$, and stabilizers agree. Thus $X^{\mathrm{\prime }s}=X^{\mathrm{\prime }}\cap \mathbf{P}(V{)}^s$. This proves (iv) and all Veronese claims.

Transport GIT properties. The linear-action theorem F3 applies to $X^{\mathrm{\prime }}\subseteq \mathbf{P}(V)$. By steps 2.1 and 3.1 its semistable set, quotient target, and quotient morphism identify with $X^{ss}(L)$, $Y$, and $\pi$ respectively. Thus the section-ring morphism is a good quotient with the orbit-closure description of its fibres, proving (iii); the linear theorem also gives openness and $G$-stability of both loci, the stable geometric quotient, and the invariant-chart criterion. Since stability is defined by finite stabilizer and closed orbit inside the same identified semistable set, its locus agrees with $X^{\mathrm{\prime }s}$, proving (ii) and the stable-locus assertions in (iv). All these constructions use the given linearization and the cited orbit/quotient results; they construct no linearization and use no Hilbert--Mumford criterion, proving (v).

Assertions (i)-(v) are established: (i) in step 1.1, (ii)-(iv) in steps 2.1, 3.1 and 3.2, and (v) in step 4.1. The proof uses only the given linearization and the orbit/quotient suppliers; it constructs no linearization and invokes no numerical criterion.