Projective GIT from Linearized Line Bundles

Let $G$ be a complex affine algebraic group acting algebraically on a classical complex variety $X$ (Classical complex affine algebraic actions and rational modules, Classical algebraic prevarieties, regular maps, and varieties), and let $L$ be an invertible sheaf on $X$ with total space $p:L\to X$ (Invertible sheaves), the total space being obtained by gluing $U\times {\mathbb{A}}^1$ over local frames of $L$ using their invertible regular transition functions, and the projection being a morphism of locally ringed spaces (Morphisms of locally ringed spaces). Write $\sigma :G\times X\to X$ for the action.

A $G$-linearization of $L$ is an algebraic $G$-action $m:G\times L\to L$ on the total space such that

• $p(m(g,\mathrm{\ell }))=gp(\mathrm{\ell })$ for all $g\in G$, $\mathrm{\ell }\in L$, and
• for every $g\in G$ and $x\in X$ the fibre map $L_x\to L_{gx}$, $\mathrm{\ell }\mapsto g\mathrm{\ell }$, is $\mathbb{C}$-linear.

A $G$-linearized invertible sheaf is an invertible sheaf together with a linearization; the pair is written $(L,m)$. Equivalently, a linearization is an isomorphism $\phi :{\sigma }^{\ast }L\to {\mathrm{p}\mathrm{r}}_2^{\ast }L$ of sheaves on $G\times X$ satisfying the cocycle identity

${\mathrm{p}\mathrm{r}}_{23}^{\ast }\phi \circ ({\mathrm{i}\mathrm{d}}_G\times \sigma {)}^{\ast }\phi =(m_G\times {\mathrm{i}\mathrm{d}}_X{)}^{\ast }\phi ,$

where $m_G:G\times G\to G$ is the multiplication; at $(g,h,x)$ both sides map the fibre $L_{ghx}$ to $L_x$. The isomorphism $\phi$ sends a vector over $gx$ to its translate by $g^{-1}$ over $x$, so its inverse recovers the action on total spaces.

For any algebraic character $\chi :G\to {\mathbf{G}}_m={\mathbb{C}}^{\times }$ the twist $(L,m{)}_{\chi }$ multiplies the fibre action by $\chi (g)$ and is again a linearization of the same invertible sheaf. In particular linearizations are not unique, the trivial action admits the trivial linearization of ${\mathcal{O}}_X$ and its twists, and for a finite-dimensional rational $G$-module $V$ the induced action on the tautological line bundle linearizes ${\mathcal{O}}_{\mathbf{P}(V)}(-1)$, and its dual linearizes ${\mathcal{O}}_{\mathbf{P}(V)}(1)$.

The two main theorems of this page always assume that a linearization is given; no general existence of linearizations is claimed here.

Assume AC inherited from the Proj construction. Let $S={⨁}_{n\ge 0}{S}_n$ be a finitely generated graded commutative $\mathbb{C}$-algebra with $S_0=\mathbb{C}$ and every $S_n$ finite-dimensional, and suppose $S$ is generated as a $\mathbb{C}$-algebra by homogeneous elements $f_1,\dots ,f_N$ of positive degrees $d_1,\dots ,d_N$; put $L=\mathrm{lcm}(d_1,\dots ,d_N)$ (with $L=1$ when $N=0$) and $d=kL$ with $k=\max (1,N-1)$.

Then the Veronese subalgebra $S^{(d)}={⨁}_{j\ge 0}{S}_{dj}$ is generated in degree one by $S_d$: every element of $S^{(d)}$ is a polynomial in elements of the finite-dimensional space $S_d$. Consequently, if $S_d\mathrm{\ne }0$, choosing a $\mathbb{C}$-basis $g_0,\dots ,g_M$ of $S_d$ and the induced graded surjection $B=\mathbb{C}[x_0,\dots ,x_M]\to S^{(d)}$, $x_i\mapsto g_i$, the canonical isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}{S}^{(d)}$ exhibits $\mathrm{Proj}S$ as a closed subscheme of the projective space ${\mathbf{P}}_{\mathbb{C}}^M=\mathrm{Proj}B$ of finite type over $\mathbb{C}$, and $\mathcal{O}(1)$ pulls back to ${\mathcal{O}}_{S^{(d)}}(1)$.

If $S_{+}=0$, then $S=\mathbb{C}$; and if $S_d=0$, degree-one generation makes $S^{(d)}=\mathbb{C}$. In either case $\mathrm{Proj}S=\mathrm{\varnothing }$ is the closed subscheme $V_{+}(x_0)$ of ${\mathbf{P}}_{\mathbb{C}}^0$, so the conclusion holds vacuously in that case as well.

• The correcting range of $d$. The repaired statement singles out the common multiple $d=kL$ with $k=\max (1,N-1)$; the original scaffold claimed the conclusion for every common multiple of the degrees of a generating set, which is false. With $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ graded by $\mathrm{deg}{x}_1=2$, $\mathrm{deg}{x}_2=12$, $\mathrm{deg}{x}_3=15$, $\mathrm{deg}{x}_4=20$, one has $L=60$, and the monomial $x_1{x}_2^4{x}_3^2{x}_4^2$ has degree $120=2L$ although it is not a product of two monomials of degree $L$: no sub-multiset of the degrees $\{2,12,12,12,12,15,15,20,20\}$ sums to $60$. Hence $S^{(L)}$ is not generated in degree one, while the lemma supplies $k=N-1=3$ and then $S^{(3L)}$ is. (This is Deligne's classical phenomenon, quoted in the weighted-projective-space literature; the proof above is self-contained.)
• Multiples of $d$. If $A$ is a standard graded $\mathbb{C}$-algebra, then so is every Veronese $A^{(l)}$: $A_{lm}=(A_l{)}^m$. Applying this to $A=S^{(d)}$ shows that $S^{(d^{\mathrm{\prime }})}$ is generated in degree one for every multiple $d^{\mathrm{\prime }}$ of $d$, Moreover, the maximal-block and box-bound argument of steps 1.2–3.1 works for every integer $k\ge \max (1,N-1)$: its only bound on $k$ is $k\ge N-1$. Thus the conclusion holds for all sufficiently large multiples $kL$ of $L$.
• No Hilbert--Mumford input. The proof uses only the monomial combinatorics of the degrees $d_i$ and the definitions of $\mathrm{Proj}$ and its twisting sheaves; no criterion for (semi)stability is involved.

For the projective constant-function conclusions in (ii) and (iii), assume AC inherited from global regular functions projective variety; the remaining conclusions use no choice principle. Let $L$ be a $G$-linearized invertible sheaf on a classical complex $G$-variety $X$ (G-linearizations of invertible sheaves on a complex G-variety). Then:

(i) for every $n\ge 0$ the tensor power $L^{\otimes n}$ (with $L^{\otimes 0}={\mathcal{O}}_X$) carries an induced $G$-linearization, functorial in $L$, and the canonical multiplication isomorphisms $L^{\otimes r}\otimes L^{\otimes s}\to L^{\otimes (r+s)}$ are $G$-equivariant;

(ii) each space of global sections $\mathrm{\Gamma }(X,L^{\otimes n})$ carries the linear action $(g\cdot \sigma )(x)=g\sigma (g^{-1}x)$, is a rational $G$-module (Classical complex affine algebraic actions and rational modules), and restriction to a $G$-stable open subset is $G$-equivariant; the constant functions in $\mathrm{\Gamma }(X,{\mathcal{O}}_X)$ are fixed by $G$, and if $X$ is projective and irreducible then $\mathrm{\Gamma }(X,{\mathcal{O}}_X)=\mathbb{C}$;

(iii) the direct sum $R(X,L)={⨁}_{n\ge 0}\mathrm{\Gamma }(X,L^{\otimes n})$ is a graded commutative $\mathbb{C}$-algebra with $G$ acting by graded algebra automorphisms, so that $R(X,L)$ is a graded rational $G$-algebra with degree-zero part $\mathrm{\Gamma }(X,{\mathcal{O}}_X)$; if $X$ is projective and irreducible this part is $\mathbb{C}$;

(iv) for global sections $\sigma \in \mathrm{\Gamma }(X,L^{\otimes r})$ and $\tau \in \mathrm{\Gamma }(X,L^{\otimes s})$ one has $X_{\sigma \otimes \tau }=X_{\sigma }\cap X_{\tau }$, and for a $G$-invariant section $\sigma$ and every $k\ge 1$ one has $X_{{\sigma }^{\otimes k}}=X_{\sigma }$, where ${\sigma }^{\otimes k}\in \mathrm{\Gamma }(X,L^{\otimes kr})$.

• Correction at degree zero. The scaffold wrote $\mathrm{\Gamma }(X,{\mathcal{O}}_X)=\mathbb{C}$; this holds for projective irreducible $X$ (as used on this page) but fails for affine $X$, where $\mathrm{\Gamma }(X,{\mathcal{O}}_X)=\mathcal{O}(X)$ is the whole coordinate ring. The statement above records the constants and the projective case separately.
• Restriction to projective $X$ used downstream. Every consumer of this item on the page works with a projective variety $X$, where also the degree-zero part of the section ring is $\mathbb{C}$.

Assume AC inherited from the quotient suppliers. Let $G$ be a complex affine algebraic group acting algebraically on a classical complex variety $X$ (Classical complex affine algebraic actions and rational modules) and let $Y$ be a $\mathbb{C}$-scheme (A locally ringed space). For the scheme, affine and sheaf clauses below, $X$ denotes its associated reduced finite-type complex scheme, obtained by gluing spectra of its affine coordinate rings; the algebraic action is interpreted on that scheme. Its complex closed points recover the classical variety (Classical algebraic prevarieties, regular maps, and varieties, Classical k-points give closed points over an algebraically closed field). Closed invariant subsets refer to underlying closed subsets with their reduced induced schemes. A morphism $\pi :X\to Y$ of $\mathbb{C}$-schemes, that is, a morphism of locally ringed spaces commuting with the structure morphisms to $\mathrm{Spec}\mathbb{C}$ (Morphisms of locally ringed spaces), is a good quotient of the action if:

(i) $\pi$ is $G$-invariant and surjective;

(ii) $\pi$ is affine, i.e. ${\pi }^{-1}(U)$ is an affine scheme for every affine open $U\subseteq Y$;

(iii) for every open $U\subseteq Y$ the pullback ${\mathcal{O}}_Y(U)\to {\mathcal{O}}_X({\pi }^{-1}U{)}^G$ is an isomorphism onto the $G$-invariant functions;

(iv) for every closed $G$-stable $Z\subseteq X$ the image $\pi (Z)$ is closed in $Y$; and

(v) for disjoint closed $G$-stable $Z_1,Z_2\subseteq X$ one has $\pi (Z_1)\cap \pi (Z_2)=\mathrm{\varnothing }$.

It is a geometric quotient if in addition for every $y\in Y(\mathbb{C})$, the complex points of its fibre ${\pi }^{-1}(y)$ form exactly one $G(\mathbb{C})$-orbit. The fibre condition is stated on complex closed points; it does not identify all scheme points with classical points.

The good-quotient clauses and the single-orbit fibre condition are separate requirements. The projective GIT theorems below establish the good quotient on the semistable locus and guarantee a geometric quotient on the stable locus. A geometric quotient can also occur outside the stable locus, since the single-orbit fibre condition does not require finite stabilizers.

• Relation to the categorical-quotient definition. A good quotient has the categorical universal property for invariant morphisms to classical varieties viewed as their associated schemes; when the target is a classical variety, this is the notion of Categorical and geometric quotients of classical varieties. On complex closed points it gives a geometric quotient there exactly when its fibres are the $G$-orbits; both implications are proved in Good quotients are local on the target and are categorical quotients below. For affine $X$, Finite generation of invariants and the affine categorical quotient supplies the classical closed-point quotient properties. The proof of Affine chart quotients for invariant sections of a linear action additionally verifies the scheme surjectivity, affine and invariant-sheaf clauses, giving good quotients of the affine charts.
• Provenance of the notion. This is Seshadri's notion as presented by Newstead in §1.4 and used by Brion (in the form ${\mathcal{O}}_Y\cong ({\pi }_{\ast }{\mathcal{O}}_X{)}^G$) and by Hoskins in §§3.3-3.4.
• AC. The definition itself uses no choice; the axiom is inherited only through the quotient suppliers that the later theorems invoke, and every theorem that uses such a supplier carries AC explicitly.

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).

Assume AC as inherited from the projective and ample-sheaf suppliers. Let $G$ be a complex affine algebraic group acting algebraically on a complex projective variety $X$ (projective variety classical), and 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). Then there exist $m\ge 1$, a finite-dimensional rational $G$-module $V$, and a $G$-equivariant closed immersion $i:X\hookrightarrow \mathbf{P}(V)$ with $i^{\ast }\mathcal{O}(1)\cong L^{\otimes m}$ as $G$-linearized invertible sheaves. Moreover $i$ can be taken to be the morphism defined by the complete linear system $\mathrm{\mid }{L}^{\otimes m}\mathrm{\mid }$.