A Generalization of Mumford’s Geometric Invariant Theory

We generalize Mumford's construction of good quotients for reductive group actions. Replacing a single linearized invertible sheaf with a certain group of sheaves, we obtain a Geometric Invariant Theory producing not only the quasiprojective quotient spaces, but more generally all divisorial ones. As an application, we characterize in terms of the Weyl group of a maximal torus, when a proper reduc-tive group action on a smooth complex variety admits an algebraic variety as orbit space.


Introduction
Let the reductive group G act regularly on a variety X.In [19], Mumford associates to every G-linearized invertible sheaf L on X a set X ss (L) of semistable points.He proves that there is a good quotient p : X ss (L) → X ss (L)/ /G, that means p is a G-invariant affine regular map and the structure sheaf of the quotient space is the sheaf of invariants.Mumford's theory is designed for the quasiprojective category: His quotient spaces are always quasiprojective.Conversely, for connected G and smooth X, if a G-invariant open set U ⊂ X has a good quotient U → U/ /G with U/ /G quasiprojective, then U is a saturated subset of a set X ss (L) for some G-linearized invertible sheaf L on X.However, there frequently occur good quotients with a non quasiprojective quotient space; even if X is quasiaffine and G is a one dimensional torus, see e.g.[2].For X = P n or X a vector space with linear G-action, the situation is reasonably well understood, see [8] and [9].But for general X, the picture is still far from being complete.The purpose of this article is to present a general theory for good quotients with so called divisorial quotient spaces.Recall from [12] that an irreducible J. Hausen variety Y is divisorial if every y ∈ Y admits an affine neighbourhood of the form Y \ Supp(D) with an effective Cartier divisor D on Y .This is a considerable generalization of quasiprojectivity.For example, all smooth varieties are divisorial.Our approach to divisorial good quotient spaces is to replace Mumford's single invertible sheaf L with a free finitely generated group Λ of Cartier divisors on X.Then a G-linearization of such a group Λ is a certain G-sheaf structure on the graded O X -algebra A associated to Λ; for the precise definitions see Section 1.In Section 2, we associate to every G-linearized group Λ ⊂ CDiv(X) a set X ss (Λ) ⊂ X of semistable points and a set X s (Λ) ⊂ X ss (Λ) of stable points.Theorem 3.1 generalizes Mumford's result on existence of good quotients: Theorem 1.For any G-linearized group Λ of Cartier divisors, there is a good quotient X ss (Λ) → X ss (Λ)/ /G with a divisorial quotient space X ss (Λ)/ /G.
We note here that our quotient spaces are allowed to be non separated; see also the brief discussion at the end of Section 3. As in the classical situation, the restriction of the above quotient map to the set of stable points separates the orbits.In Theorem 4.1, we give a converse of the above result: Theorem 2. For Q-factorial, e.g.smooth, X every G-invariant open subset U ⊂ X with a good quotient such that U/ /G is divisorial occurs as a saturated subset of a set of semistable points X ss (Λ).
As an application, we discuss actions of connected reductive groups G on normal complex varieties X.The starting point is the reduction theorem of A. Bia lynicki-Birula and J. Świȩcicka [6,Theorem 5.1]: If some maximal torus T ⊂ G admits a good quotient X → X/ /T , then there is a "good quotient" for the action of G on X in the category of algebraic spaces.Examples show that in general, the quotient space really drops out of the category of algebraic varieties, see [7, page 15].So, there arises a natural question: When there is a good quotient X → X/ /G in the category of algebraic varieties?Our answers to this question are formulated in terms of the normalizer N (T ) of a maximal torus T ⊂ G. Recall that the connected component of the unit element of N (T ) is just T ; in other words N (T )/T is finite.The first result is the following, see Theorem 5.1: Theorem 3. Let G be a connected reductive group, and let X be a normal complex G-variety.Then the following statements are equivalent: i) There is a good quotient X → X/ /G with a divisorial prevariety X/ /G.ii) There is a good quotient X → X/ /N (T ) with a divisorial prevariety X/ /N (T ).
Moreover, if one of these statements holds with a separated quotient space then so does the other.
We specialize to proper G-actions.It is an easy consequence of the reduction theorem [6, Theorem 5.1] that such an action always admits a "geometric quotient" in the category of algebraic spaces.Fundamental results of Kollár [18], Keel and Mori [15] extend this fact to a more general framework.
In our second result, Theorem 5.2, the words geometric quotient refer to a good quotient (in the category of algebraic varieties) that separates orbits: Theorem 4. Suppose that a connected reductive group G acts properly on a Q-factorial complex variety X.Then the following statements are equivalent: i) There exists a geometric quotient X → X/G.
ii) There exists a geometric quotient X → X/N (T ).
Moreover, if one of these statements holds, then the quotient spaces X/G and X/N (T ) are separated and Q-factorial.
So, for proper G-actions on Q-factorial varieties, the answer to the above question is encoded in an action of the Weyl group W := N (T )/T : A geometric quotient X → X/G exists in the category of algebraic varieties if and only if the induced action of W on X/T admits an algebraic variety as orbit space.

G-linearization and ample groups
Throughout the whole article, we work in the category of algebraic prevarieties over an algebraically closed field K.In particular, the word point refers to a closed point.First we fix the notions concerning group actions and quotients.
In this section, G denotes a linear algebraic group, and X is an irreducible Gprevariety, that means X is an irreducible (possibly non separated) prevariety (over K) together with a regular group action σ : G × X → X.
For reductive G, a good quotient of the G-prevariety X is a G-invariant affine regular map p : X → X/ /G of prevarieties such that p * : O X/ /G → p * (O X ) G is an isomorphism.By a geometric quotient we mean a good quotient that separates orbits.Geometric quotient spaces are denoted by X/G.
Remark 1.1.[22,Theorem 1.1].Let p : X → X/ /G be a good quotient for an action of a reductive group G. Then we have: Now we introduce the basic concepts used in this article, compare also [13] and [14].When we speak of a subgroup of the group CDiv(X) of Cartier divisors of X, we always mean a finitely generated free subgroup.

J. Hausen
Let Λ ⊂ CDiv(X) be such a subgroup.Denoting by A D := O X (D) the sheaf of sections of D ∈ Λ, we obtain a Λ-graded O X -algebra: The following notion extends Mumford's concept of a G-linearized invertible sheaf to groups of divisors: The reason to introduce besides the straightforward generalization 1.2 ii) also the weaker notion 1.2 i), is that in practice the latter is often much easier to handle.However, in many important cases both notions coincide, for example if the component G 0 of the unit element is a torus: Proof.Assume that Λ ⊂ CDiv(X) is G-linearized, and let A D be a homogeneous component of the associated graded O X -algebra.Consider a geometric line bundle p : L → X having A D as its sheaf of sections.Then the G-sheaf structure of A D gives rise to a set theoretical action, namely where for given z ∈ L we choose any local section f of A D satisfying f (p(z)) = z.Note that this well defined.In view of [16,Lemma 2.3], we only have to show that this action is regular.Since for fixed g ∈ G the map z → g • z is obviously regular, it suffices to show that G 0 × L → L is regular.
According to our assumption on X, it suffices to treat the case that X is affine.But then the rational representation of G 0 on the O(X)-algebra defines a regular G 0 -action on the dual bundle L := Spec(A) such that L → X becomes equivariant and G 0 acts linearly on the fibres.It is straightforward to check that this G 0 -action on L is dual to the G 0 -action on L. Hence also the latter action is regular.Proof.To distinguish the two G-sheaf structures on the graded O X -algebra associated to Λ, we denote them by (g, f ) → g •f and (g, f ) → g * f .Consider a homogeneous component A D , and the tensor product Since as an O X -module, A D ⊗ OX A −D is isomorphic to the structure sheaf itself, we obtain a G-sheaf structure on O X , also denoted by (g, f ) → g • f .As it arises from a G-linearization in the sense of [19,Definition 1.6], this G-sheaf structure is of the form with a function χ ∈ O * (G × X).Since we assumed O * (X) ∼ = K * , the function χ does not depend on the second variable.In fact, χ even turns out to be a character on G. Now, replacing in this setting D with a multiple nD amounts to replacing χ with χ n .Thus, taking n to be the order of the character group of G, we see that for any D ∈ Λ, the two G-sheaf structures on A nD coincide.The assertion follows.
We look a bit closer to the O X -algebra A associated to a group Λ ⊂ CDiv(X).This algebra gives rise to a prevariety X := Spec(A) and a canonical map q : X → X.We list some basic features of this construction: Remark 1.6.Let X := Spec(A) and q : X → X be as above.For an open subset U ⊂ X, set U := q −1 (U ).

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J. Hausen i) For a section f ∈ A D (U ), let Z(f ) := Supp(D| U + div(f )) denote the set of its zeroes.Then we have ii) The algebraic torus H := Spec(K[Λ]) acts regularly on X such that every f ∈ A D (U ) is homogeneous with respect to the character χ D , i.e., we have iii) The action of H on X is free and the map q : X → X is a geometric quotient for this action.
For the subsequent constructions, it is important to figure out those groups Λ ⊂ CDiv(X) for which the associated prevariety X over X is in fact a quasiaffine variety.This leads to the following notion: Definition 1.7.We call the group Λ ⊂ CDiv(X) ample on an open subset U ⊂ X, if there are homogeneous sections f 1 , . . ., f r ∈ A(U ) such that the sets U \ Z(f i ) are affine and cover U .
If Λ ⊂ CDiv(X) is ample on X, then we say for short that Λ is ample.So, the prevariety X admits an ample group Λ ⊂ CDiv(X) if and only if it is divisorial in the sense of Borelli [12], i.e., every x ∈ X has an affine neighbourhood X \ Supp(D) with some effective D ∈ CDiv(X).
Remark 1.8.If X is a divisorial prevariety, then the intersection U ∩ U of any two affine open subsets U, U ⊂ X is again affine.
In the following statement, we subsume the consequences of the existence of a G-linearized ample group, compare [13,Section 2].By an affine closure of a quasiaffine variety Y we mean an affine variety Y containing Y as an open dense subvariety.
Proposition 1.9.Let G be a linear algebraic group and let X be a Gprevariety.Suppose that Λ ⊂ CDiv(X) is G-linearized and ample on some where q : X → X is as above.
i) U is quasiaffine and the representation of G on O( U ) induces a regular G-action on U such that the actions of G and H := Spec(K[Λ]) commute and the canonical map q : U → X becomes G-equivariant.ii) For any collection f 1 , . . ., f r ∈ A(U ) satisfying the ampleness condition, there exists a (G × H)-equivariant affine closure U of U such that the f i extend to regular functions on U and q −1 (U fi ) = U fi holds.
Proof Let G be a reductive algebraic group, and let X be an irreducible G-prevariety.
Suppose that Λ ⊂ CDiv(X) is a G-linearized (finitely generated free) subgroup.Denote the associated Λ-graded O X -algebra by Definition 2.1.Let G, X, Λ and A be as above.We say that a point Following Mumford's notation, we denote the G-invariant open sets corresponding to the semistable, stable and properly stable points by X ss (Λ), X s (Λ) and X s 0 (Λ) respectively.If we want to specify the acting group G, we also write X ss (Λ, G) etc.. Remark 2.2.Let X be complete, let D ∈ CDiv(X) be an effective Cartier divisor and suppose that the invertible sheaf L := A D on X is G-linearized in the sense of [19,Definition 1.6].Then the induced G-sheaf structure of A D extends to a G-linearization of Λ := ZD.Moreover, i) X ss (Λ) contains precisely the points of X which are semistable in the sense of [19, Definition 1.7 i)], ii) X s (Λ) contains precisely the points of X which are stable in the sense of [19, Definition 1.7 ii)], iii) X s 0 (Λ) contains precisely the points of X which are properly stable in the sense of [19,Definition 1.8].
The remainder of this section is devoted to giving a geometric interpretation of semistability.For this, let U ⊂ X denote any G-invariant open subset such that Λ is ample on U and X ss (Λ) is contained in U , for example U = X ss (Λ).As usual, let Documenta Mathematica 6 (2001) 571-592

J. Hausen
Recall from Section 1 that the map q : X → X is a geometric quotient for the action of H := Spec(K[Λ]) on X induced by the Λ-grading of A. Moreover, U is a quasiaffine variety and carries a regular G-action making q : U → X equivariant.
Our description involves two choices.First let f 1 , . . ., f r ∈ A(X) be homogeneous G-invariant sections such that the sets X \Z(f i ) are as in Definition 2.1 i) and cover X ss (Λ).
Next choose a (G×H)-equivariant affine closure U of U such that the functions f i ∈ O( U ) extend regularly to U and U fi = U fi holds for each i = 1, . . ., r.
Consider the good quotient Then the quotient variety U / /G inherits a regular action of H such that the map p : U → U / /G becomes H-equivariant.In this setting, the set U \ q −1 (X ss (Λ)) takes over the role of the classical nullcone: The main point in the proof is to express Condition 2. Proof of Proposition 2.3.Let W := X ss (Λ) and W := q −1 (W ).We begin with the inclusion "⊂" of assertions i) and ii).First note that W is p-saturated, because this holds for each U fi and, according to Remark 1.6 i), W is covered by these subsets.In particular, it follows p( W ) ⊂ V 0 .
To verify p( W . By the properties of f i , the function h satisfies the condition of Lemma 2.4 for the point p(z).Hence H p(z) is finite, which means p(z) We come to the inclusion "⊃" of assertion ii).Let y provides an h ∈ O(X/ /G), homogeneous with respect to some χ D ∈ Char(H), such that y ∈ V := (X/ /G) h holds and the D ∈ Λ admitting an invertible χ D -homogeneous function on V form a subgroup of finite index in Λ. Suitably modifying h, we achieve additionally Now, consider a point z ∈ p −1 (y).Since y ∈ V 0 , we have z ∈ X.We have to show that q(z) is semistable.For this, consider the G-invariant homogeneous section f := p * (h)| X of A D (X).By the choice of h, this f fulfills the conditions of Definition 2.1 i) and thus the point q(z) is in fact semistable.
i) A point x ∈ X ss (Λ) with an orbit G•x of maximal dimension is stable if and only if for any z ∈ q −1 (x) the orbit G•z is closed in X. ii) A point x ∈ X ss (Λ) with finite isotropy group G x is properly stable if and only if for any z ∈ q −1 (x) the orbit G•z is closed in X.

The quotient of the set of semistable points
Let G be a reductive algebraic group, and let X be a G-prevariety.In this section we show that any set of semistable points admits a good quotient.The result generalizes [19, Theorem 1.10].
An immediate consequence of this result is that the set of stable points admits a geometric quotient.More precisely, by the properties of good quotients we have: Remark 3.2.In the notation of 3.1, the set X s (Λ) is p-saturated and the restriction p : X s (Λ) → p(X s (Λ)) is a geometric quotient.
In the proof of Theorem 3.1, we make use of the following observation on geometric quotients for torus actions, compare [ Each T -invariant and hence we have h i = p * (h i ) with a regular function h i defined on V i := p(U i ).By construction, the zero set of h i is just B ∩ V i .Since every h i /h j is regular and invertible on V i ∩ V j , the functions h i yield local equations for an effective Cartier divisor E on Z having support B.
Proof of Theorem 3.1.As usual, let A be the graded O X -algebra associated to Λ.We consider the corresponding prevariety X := Spec(A) and the map q : X → X. Recall that the latter is a geometric quotient for the action of H := Spec(K[Λ]) on X. Set for short W := X ss (Λ).Surely, Λ is ample on W .
Proposition 1.9 yields that W := q −1 (W ) is a quasiaffine variety.Moreover, W carries a G-action that commutes with the action of H and makes q : W → W equivariant.Choose f 1 , . . ., f r ∈ A(X) satisfying the conditions of Definition 2.1 such that W is covered by the affine sets X \ Z(f i ), and set Choose a (G × H)-equivariant affine closure W of W such that the above Now it is straightforward to check that the induced map W → ( W / /G)/H is the desired good quotient for the action of G on W .
We conclude this section with a short discussion of the question, when the quotient space X ss (Λ)/ /G is separated.Translating the usual criterion for separateness in terms on functions on the quotient space to the setting of invariant sections of the O X -algebra A of a G-linearized group Λ, we obtain: Remark 3.4.Let Λ ⊂ CDiv(X) be a G-linearized group on a G-variety X, and let X ss (Λ) be covered by X \Z(f i ) with G-invariant sections f 1 , . . ., f r ∈ A(X) as in 2.1 i).The quotient space X ss / /G is separated if and only if for any two indices i, j the multiplication map defines a surjection in degree zero: In the classical setting [19, Definition 1.7], the group Λ is of rank one, and the above sections f i are of positive degree.In particular, for suitable positive powers n i , all sections f ni i are of the same degree, and Remark 3.4 implies that the resulting quotient space is always separated.
As soon as we leave the classical setting, the above reasoning may fail, and we can obtain nonseparated quotient spaces, as the following two simple examples show.Both examples arise from the hyperbolic K * -action on the affine plane.
In the first one we present a group Λ of rank one defining a nonseparated quotient space: Example 3.5.Let the onedimensional torus T := K * act diagonally on the punctured affine plane Consider the group Λ ⊂ CDiv(X) generated by the principal divisor D := div(z 1 ).Since D is T -invariant, the group Λ is canonically T -linearized.We claim that the corresponding set of semistable points is To verify this claim, let A denote the graded O X -algebra associated to Λ, and consider the T -invariant sections Then the sets X \ Z(f 1 ) and X \ Z(f 2 ) form an affine cover of X.Moreover, we have T -invariant invertible sections: So, f 1 , f 2 ∈ A(X) satisfy the conditions of Definition 2.1 i), and the claim is verified.The quotient space Y := X ss (Λ)/ /T is the affine line with doubled zero.In particular, Y is a nonseparated prevariety.
In view of Remark 3.4, we obtain always separated quotient spaces when starting with a group Λ = ZD, where D is a divisor on a complete G-variety X.
In this setting, the lack of enough invariant sections of degree zero on the sets X \ Z(f i ) occurs for groups Λ of higher rank: Example 3.6.Let the onedimensional torus T := K * act diagonally on the projective plane Consider the group Λ ⊂ CDiv(X) generated by the divisors , where E i denotes the prime divisor V (X; z i ).Since the divisors D i are T -invariant, the group Λ is canonically T -linearized.We claim that the corresponding set of semistable points is To check this claim, denote the right hand side by U .Let A again denote the graded O X -algebra associated to Λ, and consider the T -invariant sections For the respective zero sets of these sections we have So, the set U is indeed the union of the affine sets X \ Z(f i ).Moreover, we have invertible sections Thus f 1 , f 2 , f 3 ∈ A(X) satisfy the conditions of Definition 2.1 i).Since the fixed points [1, 0, 0], [0, 1, 0] and [0, 0, 1] occur as limit points of suitable Torbits through U , they cannot be semistable.The claim is verified.Note that X ss (Λ) equals in fact the set of (properly) stable points.The quotient space Y := X ss (Λ)/ /T is a projective line with doubled zero.In particular, Y is a nonseparated prevariety.

Good quotients for Q-factorial G-varieties
Let G be a not necessarily connected reductive group, and let X be an irreducible G-prevariety.In [19, Converse 1.13], Mumford shows that, provided X is a smooth variety and G is connected, every open subset U with a geometric quotient U → U/G such that U/G is quasiprojective arises in fact from a set of stable points.
Here we generalize this statement to non connected G and open subsets with a divisorial good quotient space.Assume that X is Q-factorial, i.e., X is normal and for each Weil divisor D on X, some multiple of D is Cartier.Moreover, suppose that X is of affine intersection, i.e., for any two open affine subsets of X their intersection is again affine.To formulate our result, let U ⊂ X be an open G-invariant set of the Gprevariety X such that there exists a good quotient U → U/ /G.Then we have: and is saturated with respect to the quotient map X ss (Λ) → X ss (Λ)/ /G.
For the proof of this statement, we need two lemmas.The first one is an existence statement on canonical linearizations: Let H be any linear algebraic group.We say that a Weil divisor E on a normal H-prevariety Y is H-tame, if Supp(E) is H-invariant and for any two prime cycles Lemma 4.2.Let Λ ⊂ CDiv(Y ) be a group consisting of H-tame divisors.Then Λ admits a canonical H-linearization, namely Proof.First we note that the canonical action of H on K(Y ) induces indeed a H-sheaf structure on the sheaf A E of an H-tame Cartier divisor E on Y .This follows from the fact that for f ∈ K(Y ), the order of a translate h•f along a prime divisor E 0 of E is given by We still have to show that for every by finitely many open affine subsets V ij ⊂ V i with the following properties: , where n i ∈ N,

J. Hausen
ii) for each k = 1, . . ., r there exists an Proof.Let y ∈ V i and consider an affine open neighbourhood V ⊂ V i of y such that on V we have E k = div(h k ) with some h k ∈ O(V ) for all k.Then each without zeroes in V .By suitably shrinking V , we achieve V = X \ Z(h) with some h ∈ B niEi (X) and some n i ∈ N. Since finitely many of such V cover V i , the assertion follows.
Proof of Theorem 4.1.Since the quotient space Y := U/ /G is divisorial, we find effective E 1 , . . ., E r ∈ CDiv(Y ) such that the sets V i := Y \ Supp(E i ) are affine and cover Y .Let V ij , h ij and h ijk as in Lemma 4.3.Consider the quotient map p : U → Y and the pullback divisors Then every U i := p −1 (V i ) is affine and equals U \ Supp(D i ).Moreover, since they are locally defined by invariant functions, we see that the divisors D i are G-tame.Since X is Q-factorial and of affine intersection, we can construct G-tame effective divisors D i ∈ CDiv(X) with the following properties: Let Λ ⊂ CDiv(X) denote the group generated by the divisors D 1 , . . ., D r , and let A be the associated graded O X -algebra.Lemma 4.2 tells us that the group Λ is canonically G-linearized by setting g•f (x) := f (g −1 •x) on the homogeneous components of A. Note that the set U ⊂ X is covered by the affine open subsets U ij := p −1 (V ij ).Thus, using the pullback data f ij and , it is straightforward to check U ⊂ X ss (Λ).Moreover, since the U ij are defined by the G-invariant sections f ij , we see that they are saturated with respect to the quotient map p : X ss (Λ) → X ss (Λ)/ /G.Hence U is p -saturated in X ss (Λ).
Corollary 4.4.Let the algebraic torus T act effectively and regularly on a Q-factorial variety X, and let U ⊂ X be the union of all T -orbits with finite isotropy group.If dim(X \ U ) < dim(T ), then U is the set of semistable points of a T -linearized group Λ ⊂ CDiv(X) .
Proof.By [23,Corollary 3], there is a geometric quotient U → U/T .Using Proposition 1.9 and Lemma 3.3, we see that U/T is a divisorial prevariety.Theorem 4.1 provides a T -linearized group Λ ⊂ CDiv(X) such that X ss (Λ) contains U as a saturated subset with respect to p : X ss (Λ) → X ss (Λ)/T .
The classical example of a generic C * -action on the Grassmannian of two dimensional planes in C 4 , compare also [5] and [25], fits into the setting of the above observation: Example 4.5.Realize the complex Grassmannian X := G(2; 4) via Plücker relations as a quadric hypersurface in the complex projective space P 5 : This allows us to define a regular action of the one dimensional torus T = C * on X in terms of coordinates: This T -action has six fixed points.Let U ⊂ X be the complement of the fixed point set.It is well known that the quotient space Y := U/T is a nonseparated prevariety which is covered by four projective open subsets.Moreover, Y contains two nonprojective complete open subsets, see [5,Remark 1.6] and [25,Example 6.4].According to Corollary 4.4, the set U can be realized as the set of semistable points of a T -linearized group of divisors.Let us do this explicitly.Consider for example the prime divisors D 1 := V (X; z 1 ) and D 2 := V (X; z 4 ) and the group Λ := ZD 1 ⊕ ZD 2 ⊂ CDiv(X).
Then the group Λ is canonically T -linearized.We show that X ss (Λ) = U holds.Let A denote the graded O X -algebra associated to Λ, and consider the following T -invariant sections f ij ∈ A(X): Hausen By definition, we have Z(f ij ) = V (X; z i z j ) for the set of zeroes of f ij .Consequently, U is the union of the affine open subsets X ij := X \ Z(f ij ).Moreover, every h ij is invertible over X ij , and the claim follows.
In fact, using Pic T (X) ∼ = Z 2 , it is not hard to show that besides the T -invariant open subsets W ⊂ X admitting a projective quotient variety W/ /T , the subset U is the only open subset of the form X ss (Λ) with a T -linearized group Λ ⊂ CDiv(X).

Reduction theorems for good quotients
In this section, G is a connected reductive group and the field K is of characteristic zero.Fix a maximal torus T ⊂ G and denote by N (T ) its normalizer in G.The first result of this section relates existence of a good quotient by G to existence of a good quotient by N (T ): Theorem 5.1.For a normal G-prevariety X, the following statements are equivalent: i) There is a good quotient X → X/ /G with a divisorial prevariety X/ /G.ii) There is a good quotient X → X/ /N (T ) with a divisorial prevariety X/ /N (T ).
Moreover, if one of these statements holds with a separated quotient space, then so does the other.
Note that if X admits a divisorial good quotient space, then X itself is divisorial.In the second result, we specialize to geometric quotients.Recall that an action of G on X is said to be proper, if the map G × X → X × X sending (g, x) to (g•x, x) is proper.
Theorem 5.2.Suppose that G acts properly on a Q-factorial variety X.Then the following statements are equivalent: i) There exists a geometric quotient X → X/G.
Moreover, if one of these statements holds, then the quotient spaces X/G and X/N (T ) are separated Q-factorial varieties.
As an immediate consequence, we obtain the following statement on orbit spaces by the special linear group SL 2 (K), which applies for example to the problem of moduli for n ordered points on the projective line, compare [20] and [4, Section 5]: Corollary 5.3.Let SL 2 (K) act properly on an open subset U ⊂ X of a Qfactorial toric variety X such that some maximal torus T ⊂ SL 2 (K) acts by means of a homomorphism T → T X to the big torus T X ⊂ X.Then there is a geometric quotient U → U/SL 2 (K).
Proof.Since SL 2 (K) acts properly, there is a geometric quotient U → U/T .Let U ⊂ X be a maximal open subset such that U ⊂ U and there is a geometric quotient U → U /T .Then the set U is invariant under the big torus T X , see e.To prove the converse, we first reduce to the case that G is semisimple: Let R ⊂ G be the radical of G. Then R is a torus, and we have R ⊂ T .In particular, there is a good quotient X → X for the action of R on X.Thus Let g ∈ G with gT g −1 = T .Then the closure of T•g•z contains a point z ∈ G•z .Surely, p T (g•z) equals p T (z ).Thus, since z ∈ X \ X, Proposition 2.3 ii) tells us that g•x = q(g•z) is not semistable with respect to T .A contradiction.
As the situation y ∈ p G (X \ X) is excluded, the isotropy group H y is of positive dimension, and the whole fibre p −1 G (y) is contained in X.Let H 0 ⊂ H y be the connected component of the neutral element.Then H 0 acts freely on the fibre p −1 G (y), and the closed orbit G•z ⊂ p −1 G (y) is invariant by H 0 .Let µ : g → g•z denote the orbit map.Since the actions of G and H 0 commute, G := µ −1 (H 0 •z ) is a subgroup of G. Since H 0 •z ∼ = H 0 , there is a torus S ⊂ G with µ(S ) = H 0 •z , use for example [11,Proposition IV.11.20].Let T ⊂ G be a maximal torus with S ⊂ T and choose g ∈ G with T = gT g −1 .Then H 0 •g•z equals (gS g −1 )•g•z .According to Proposition 2.3 ii), the point q(g•z ) is not semistable with respect to T .A contradiction.So, every x ∈ X is semistable with respect to G, and the implication "ii)⇒i)" is proved.We come to the supplement concerning separateness.Clearly, existence of a good quotient X → X/ /G with X/ /G separated implies that also the quotient space X/ /N (T ) is separated.For the converse, suppose that X → X/ /N (T ) exists with a separated divisorial X/ /N (T ).Then there is a good quotient X → X/ /T with a separated quotient space X/ /T , and [6,Theorem 5.4] implies that also the quotient space X/ /G is separated.
In the proof of Theorem 5.2, we shall use that geometric quotient spaces of proper actions inherit Q-factoriality.By the lack of a reference for this presumably well-known fact, we give here a proof: Lemma 5.5.Suppose that a reductive group H acts regularly with finite isotropy groups on a variety Y and that there is a geometric quotient p : Y → Y /H.If Y is Q-factorial, then so is Y /H.
Proof.Assume that Y is Q-factorial, and let E ⊂ Y /H be a prime divisor.Then p −1 (E) is a union of prime divisors D 1 , . . ., D r .Some multiple mD of the divisor D := D 1 +. ..+D r is Cartier.Using Lemma 4.2 and Proposition 1.3, we see that the group of Cartier divisors generated by mD is canonically strongly H-linearized.Enlarging m, we achieve that the sheaf A mD is equivariantly isomorphic to the pullback p * (L) of some invertible sheaf L on Y /H, use e.g.[17,Proposition 4.2].The canonical section 1 ∈ A mD (Y ) is H-invariant and hence induces a section f ∈ L(Y /H) having precisely E as its set of zeroes.
Proof of Theorem 5.2.If one of the quotients exists, then by [19, Section 0.4] and Lemma 5.5, the quotient space is separated and Q-factorial.Now, existence of a geometric quotient X → X/G surely implies existence of a geometric quotient X/N (T ).Conversely, if X/N (T ) exists, then it is Q-factorial.Hence Theorem 5.1 yields a geometric quotient X → X/G.Documenta Mathematica 6 (2001) 571-592 1 i) in terms of the action of the torus H on the affine variety U / /G.Consider more generally an arbitrary algebraic torus T and a quasiaffine T -variety Y .Lemma 2.4.The isotropy group T y of a point y ∈ Y is finite if and only if there is a homogeneous function h ∈ O(Y ) such that Y h is an affine neighbourhood of y and the characters χ ∈ Char(T ) admitting an invertible χ -homogeneous h ∈ O(Y h ) form a sublattice of finite index in Char(T ).Proof.First suppose that T y is finite.Consider the orbit B := T •y.This is a locally closed affine subvariety of Y .The set M consisting of all characters χ ∈ Char(T ) admitting a χ -homogeneous h ∈ O(B) with h (y) = 1 is a sublattice of Char(T ).We show that M is of full rank: Otherwise there is a non trivial one parameter subgroup λ : K * → T such that χ•λ = 1 holds for every χ ∈ M .Thus, by the definition of M , all homogeneous functions of O(B) are constant along λ(K * )•y.As these functions separate the points of B, we conclude λ(K * ) ⊂ T y .A contradiction.Now, choose any T -homogeneous function h ∈ O(Y ) such that Y h is affine, contains B as a closed subset, and for some base χ 1 , . . ., χ d of M the associated functions h i ∈ O(B) extend to invertible regular homogeneous functions on Y h .Then this h ∈ O(Y ) is as desired.The "if" part of the assertion is settled by similar arguments.

Lemma 3 . 3 .
Let T be an algebraic torus and suppose that Y is an irreducible quasiaffine T -variety with geometric quotient p : Y → Y /T .Then Y /T is a divisorial prevariety.Proof.We may assume that T acts effectively.Set for short Z := Y /T .Given a point z ∈ Z, choose a T -homogeneous f ∈ O(Y ) such that U := Y f is an affine neighbourhood of p −1 (z).Consider the affine neighbourhood V := p(U ) of z.We show that B := Z \ V is the support of an effective Cartier divisor on Y .Let χ ∈ Char(T ) be the weight of the above f ∈ O(Y ).Since T acts effectively with geometric quotient, all isotropy groups T y are finite.So we can use Lemma 2.4 to cover Y by T -invariant affine open sets U i admitting invertible functions g i ∈ O(U i ) that are homogeneous with respect to some common multiple mχ.
and A E (V ) injects H-equivariantly into O(V ).Hence [16, Section 2.5] gives the claim.Now, consider a normal prevariety Y with effective E 1 , . . ., E r ∈ CDiv(Y ) such that the sets V i := Y \ Supp(E i ) are affine and cover Y .Let Γ ⊂ CDiv(Y ) be the subgroup generated by E 1 , . . ., E r .Denote the associated Γ-graded O Y -algebra by B := E∈Γ B E := E∈Γ O Y (E).

Lemma 4 . 3 .
In the above setting, every open set g.[24, Corollary 2.4].Thus the geometric quotient space Y := U /T is again a toric variety.In particular, any two points y, y ∈ Y admit a common affine neighbourhood in Y .But this property is inherited by Y := U/T .Thus, since W := N (T )/T is of order two, we obtain a geometric quotient Y → Y /W .The composition of U → Y and Y → Y /W is a geometric quotient for the action of N (T ) on U .So Theorem 5.2 gives the claim.We come to the proof of Theorems 5.1 and 5.2.We make use of the following well known fact on semisimple groups: Lemma 5.4.If G is semisimple then the character group of N (T ) is finite.Proof.It suffices to show that for each χ ∈ Char(N (T )), the restriction χ := χ| T is trivial.Clearly χ is fixed under the action of the Weyl group W = N (T )/T on R ⊗ Z Char(T ) induced by the N (T )-action(n•α)(t) := α(n −1 tn)on Char(T ).On the other hand, W acts transitively on the set of Weyl chambers associated to the root system determined by T ⊂ G. Consequently, χ lies in the closure of every Weyl chamber and hence is trivial.Proof of Theorem 5.1.The implication "i)⇒ii)" is easy, use[21, Lemma 4.1].
holds and G has only finitely many characters, then the two G-linearizations coincide on a subgroup Λ ⊂ Λ of finite index.