Projective GIT quotient for a linear action

The invariant ring and the target. If $X=\mathrm{\varnothing }$, its homogeneous coordinate ring is zero, the invariant ring and every localized chart ring are zero, both loci and every quotient target are empty, and all conclusions hold with the empty morphisms. Assume henceforth $X\mathrm{\ne }\mathrm{\varnothing }$. By [F1] $R^G$ is a finitely generated graded $\mathbb{C}$-algebra with $(R^G{)}_0=\mathbb{C}$ and finite-dimensional graded pieces; by [F2] the scheme $Y=\mathrm{Proj}{R}^G$ is projective of finite type over $\mathbb{C}$. This is assertion (i).

The semistable locus. By definition $x\in X^{ss}$ exactly when $f(x)\mathrm{\ne }0$ for some homogeneous invariant $f$ of positive degree, i.e. exactly when $x$ lies in one of the charts $X_f$ with $f\in R_{>0}^G$; each such chart is open, affine and $G$-stable by [F3], so $X^{ss}$ is open and $G$-stable and covered by the charts $X_f$.

Gluing the chart morphisms. The chart morphisms ${\pi }_f:X_f\to D_{+}(f)$ of [F3] agree on overlaps $X_{fg}$ by the compatibility assertion of [F3]; since the $D_{+}(f)$ with $f\in R_{>0}^G$ cover $Y$ and the $X_f$ cover $X^{ss}$, they glue to a $G$-invariant morphism $\pi :X^{ss}\to Y$ whose restrictions are the good quotients ${\pi }_f$. For any positive-degree invariants $f,g$, the function $f^{\mathrm{deg}g}\mathrm{/}{g}^{\mathrm{deg}f}$ on $X_g$ is the pullback of the same fraction on $D_{+}(g)$; it is nonzero precisely on $X_f\cap X_g$ and on $D_{+}(f)\cap D_{+}(g)$, respectively. Thus ${\pi }^{-1}(D_{+}(f))=X_f$, since the $X_g$ cover $X^{ss}$, so the chart maps really are target restrictions of $\pi$. By [F4] the good-quotient property is local on the target, so $\pi$ is a good quotient; in particular by [F6] it is surjective, its pullback identifies ${\mathcal{O}}_Y$ with $({\pi }_{\ast }{\mathcal{O}}_{X^{ss}}{)}^G$, and images of closed $G$-stable (respectively disjoint closed $G$-stable) subsets are closed (respectively disjoint).

Closed charts and invariant vanishing. Call a chart $X_f$, $f\in R_{>0}^G$, closed if every $G$-orbit contained in $X_f$ is closed in $X_f$, and let $X_c$ be the union of the closed charts; this is an open $G$-stable subset of $X^{ss}$. If $X_f$ is closed and $g\in R_{>0}^G$, then $X_{fg}=X_f\cap X_g$ is an open $G$-stable subset of $X_f$, so every orbit contained in $X_{fg}$ is closed in $X_{fg}$ as well. For a homogeneous invariant section $F$, if $F(x)=0$ then its closed zero locus is $G$-stable and contains $Gx$, hence also contains every $z\in \overline{Gx}\cap X^{ss}$. Equivalently, if such a $z$ lies in a chart $X_f$ and $F=f$, then $F(x)\mathrm{\ne }0$, so $x\in X_f$.

The stable locus lies in the closed charts. Let $x\in X^s$ and choose $f_0\in R_{>0}^G$ with $x\in X_{f_0}$ (step 1.2). The set $Z=\{y\in X_{f_0}:\dim G_y>0\}$ is closed and $G$-stable in $X_{f_0}$ by [F5] and is disjoint from the closed orbit $Gx$, whose stabilizers are conjugate to the finite group $G_x$. Since $Gx$ and $Z$ are disjoint closed $G$-stable subsets of the affine chart, the good-quotient property [F4] gives ${\pi }_{f_0}(Gx)\cap {\pi }_{f_0}(Z)=\mathrm{\varnothing }$. Thus ${\pi }_{f_0}(x)\mathrm{\notin }{\pi }_{f_0}(Z)$, and the affine quotient separation lemma [F7] gives $h\in \mathcal{O}(X_{f_0}{)}^G$ with $h(x)\mathrm{\ne }0$ and $h{\mathrm{\mid }}_Z=0$. By [F3] write $h=g\mathrm{/}{f}_0^m$ with $g\in R^G$ homogeneous. Choose $N\ge 1$ and put $F=g{f}_0^N$; then $F$ is homogeneous invariant of positive degree and $F(x)\mathrm{\ne }0$. Its chart satisfies $X_F\subseteq X_{f_0}$, and if $y\in X_F$ then $h(y)=g(y)\mathrm{/}{f}_0(y{)}^m\mathrm{\ne }0$, so $y\mathrm{\notin }Z$; hence every point of $X_F$ has finite stabilizer. If an orbit $Gy\subseteq X_F$ were not closed in $X_F$, its boundary in $X_F$ would contain a point $z\in X_F\cap \overline{Gy}\setminus Gy$; by [F5] the orbit $Gz$ has dimension strictly smaller than $\dim Gy$. But every point of $X_F$ has finite stabilizer, so every orbit in $X_F$ has dimension $\dim G$ by the orbit-stabilizer formula [F5], a contradiction. Thus $X_F$ is a closed chart containing $x$. Every stable point lies in such a chart, so $X^s\subseteq X_c$.

Points of closed charts with finite stabilizer are stable. Let $x\in X_c$ with $G_x$ finite and choose a closed chart $X_f$ containing $x$. The fibre ${\pi }^{-1}(\pi (x))$ is closed in $X^{ss}$ and contains $Gx$, so it contains $\overline{Gx}\cap X^{ss}$. Since $\pi (x)\in D_{+}(f)$ and $X_f={\pi }^{-1}(D_{+}(f))$ by step 2.1, the fibre lies in $X_f$. Thus every $z\in \overline{Gx}\cap X^{ss}$ lies in the closure of $Gx$ computed in $X_f$, and closedness of the orbit in that chart forces $z\in Gx$. Hence $Gx$ is closed in $X^{ss}$ and $x\in X^s$. With step 2.3 this gives $X^s=X_c\cap \{x\in X:\dim G_x=0\}$.

Fibres of $\pi$. Let $x_1,x_2\in X^{ss}$. If $\pi (x_1)=\pi (x_2)$, choose $f\in R_{>0}^G$ with $\pi (x_1)\in D_{+}(f)$; then $x_1,x_2\in X_f$ and, since ${\pi }_f$ restricts $\pi$ by step 2.1, ${\pi }_f(x_1)={\pi }_f(x_2)$, so the closures of $G{x}_1$ and $G{x}_2$ in $X_f$ meet by the unique-closed-orbit property of the affine fibre ([F4]), hence their closures in $X^{ss}$ meet. Conversely let $z\in \overline{G{x}_1}\cap \overline{G{x}_2}\cap X^{ss}$ and choose $f\in R_{>0}^G$ with $z\in X_f$; by the vanishing observation of step 2.2 the points $x_1,x_2$ also lie in $X_f$, and $z$ lies in both closures computed in $X_f$, so ${\pi }_f(x_1)={\pi }_f(z)={\pi }_f(x_2)$ by continuity and $\pi (x_1)=\pi (x_2)$. This proves (iii).

Openness and the geometric quotient. By step 3.1 the stable locus is $X^s=X_c\cap \{x\in X:\dim G_x=0\}$. The union $X_c$ of charts is open and $G$-stable, and $\{x:\dim G_x=0\}$ is open in $X_c$ by upper semicontinuity of $x\mapsto \dim G_x$ ([F5]); hence $X^s$ is open in $X$, and it is $G$-stable because stabilizers of points in one orbit are conjugate. On a closed chart $X_f$ the affine quotient ${\pi }_f$ of [F3] is a good quotient whose fibres contain a unique closed orbit ([F4]); since every orbit in $X_f$ is closed, each fibre is a single orbit, so ${\pi }_f$ is a geometric quotient. These geometric quotients agree on overlaps by [F3], so by locality of the geometric-quotient property ([F4]) they glue to a geometric quotient ${\pi }_c:X_c\to Y_c:=\pi (X_c)$, where $Y_c$ is the union of the open sets $D_{+}(f)$ over the closed charts, hence open in $Y$. The closed $G$-stable subset $X_c\setminus X^s$ is mapped by the quotient ${\pi }_c$ to a closed subset of $Y_c$ by clause (iv) of [F6], so $Y^s:=\pi (X^s)=Y_c\setminus \pi (X_c\setminus X^s)$ is open in $Y_c$ and hence in $Y$; and $X^s={\pi }^{-1}(Y^s)$ because the fibres of ${\pi }_c$ are the orbits and $X^s$ is $G$-stable. The restriction of the geometric quotient ${\pi }_c$ to the open $G$-stable subset $X^s$ is again a geometric quotient onto $Y^s$ by locality, so $\pi :X^s\to Y^s$ is a geometric quotient. This proves (iv); if $X^s=X^{ss}$ then $X_c=X^{ss}$ and the same argument shows that $\pi$ is a geometric quotient of $X^{ss}$.

Assertions (i)-(iv) are established: (i) in step 1.1, (ii) in step 1.2, (iii) in steps 2.1 and 3.2, and (iv) in steps 2.3, 3.1 and 4.1.