A Generalization of Mumford’s

Geometric Invariant Theory

Jürgen Hausen

Received: August 10, 2001

Communicated by Thomas Peternell

Abstract. 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 reductive group action on a smooth complex variety admits an algebraic variety as orbit space.

2000 Mathematics Subject Classification: 14L24, 14L30

Keywords and Phrases: Geometric Invariant Theory, good quotients, reductive group actions

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 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. Białynicki-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.

1. 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:

i) For every G-invariant closed set A ⊂ X the image p(A) ⊂ X//G is closed. ii) If A, B ⊂ X are closed G-invariant subsets, then p(A ∩ B) equals p(A) ∩

p(B). iii) Each fibre p^{−1}(y) contains exactly one closed G-orbit. iv) Every G-invariant regular map X → X′ factors uniquely through p.

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.

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:

A := ⨁_{D∈Λ} A_{D}.

The following notion extends Mumford’s concept of a G-linearized invertible sheaf to groups of divisors:

Definition 1.2. Fix the canonical G-sheaf structure (g·f)(x) := f(g^{−1}·x) on the structure sheaf O_{X}.

i) A G-linearization of the group Λ is a graded G-sheaf structure on the Λ-graded O_{X}-algebra A such that the representation of G on A(U) is rational for every G-invariant open subset U ⊂ X.

ii) A strong G-linearization of the group Λ is a G-linearization of Λ such that on each homogeneous component A_{D}, D ∈ Λ, the G-sheaf structure arises from a G-linearization σ^{∗}(A_{D}) ≅ pr_{X}^{∗}(A_{D}) in the sense of [19, Definition 1.6].

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:

Proposition 1.3. If X is covered by G^{0}-invariant affine open subsets, then every G-linearization of Λ is in fact a strong G-linearization of Λ.

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

G × L → L, (g, z) ↦ g·z := (g·f)(g·p(z)),

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

A := ⨁_{n∈ℕ} A_{nD}(X)

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.

Concerning existence of linearizations, we have the following generalization of [19, Corollary 1.6], compare [14, Proposition 3.6]:

Proposition 1.4. Suppose that G is connected and that X is a normal separated variety. Then every group Λ ⊂ CDiv(X) admits a strongly G-linearized subgroup Λ′ ⊂ Λ of finite index.

Proof. Choose a basis D_{1}, …, D_{r} of Λ. According to [16, Proposition 2.4], there is a positive integer n such that each sheaf A_{nD_{i}} admits a G-linearization in Mumford’s sense. Tensoring these linearizations gives the desired strong linearization of the subgroup Λ′ ⊂ Λ generated by nD_{1}, …, nD_{r}.

A more special existence statement for non connected G will be given in 4.2. There is also a uniqueness statement like [19, Proposition 1.4]. Note that in our version, we do not assume G to be connected:

Proposition 1.5. Suppose that Λ ⊂ CDiv(X) admits two strong G-lineariza-tions. If O^{∗}(X) = 𝕂^{∗} holds and G has only finitely many characters, then the two G-linearizations coincide on a subgroup Λ′ ⊂ Λ of finite index.

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

A_{D} ⊗_{O_{X}} A_{−D}, g • (f ⊗ h) := g·f ⊗ g∗h.

Since as an O_{X}-module, A_{D} ⊗_{O_{X}} 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

(g • f)(x) = χ(g, x)f(g^{−1}·x)

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 X̂ := Spec(A) and a canonical map q: X̂ → X. We list some basic features of this construction:

Remark 1.6. Let X̂ := Spec(A) and q: X̂ → X be as above. For an open subset U ⊂ X, set Û := q^{−1}(U). 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

Û_{f} := {x ∈ Û; f(x) ≠ 0} = q^{−1}(U \ Z(f)).

ii) The algebraic torus H := Spec(𝕂[Λ]) acts regularly on X̂ such that every f ∈ A_{D}(U) is homogeneous with respect to the character χ^{D}, i.e., we have

f(t·x) = χ^{D}(t)·f(x).

iii) The action of H on X̂ is free and the map q: X̂ → 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 X̂ 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 \overline{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 G-invariant open U ⊂ X. Let Û := q^{−1}(U) ⊂ X̂, where q: X̂ → X is as above.

i) Û is quasiaffine and the representation of G on O(Û) induces a regular G-action on Û such that the actions of G and H := Spec(𝕂[Λ]) commute and the canonical map q: Û → 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 \overline{U} of Û such that the f_{i} extend to regular functions on \overline{U} and q^{−1}(U_{f_{i}}) = \overline{U}_{f_{i}} holds.

Proof. Use [13, Lemmas 2.4 and 2.5].

2. Stability notions

Generalizing [19, Definitions 1.7 and 1.8] we shall associate to a linearized group of divisors sets of semistable, stable and properly stable points. Moreover, for ample linearized groups, we give a geometric interpretation of semistability in terms of a generalized nullcone. 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

A = ⨁_{D∈Λ} A_{D}.

Definition 2.1. Let G, X, Λ and A be as above. We say that a point x ∈ X is

i) semistable, if x has an affine neighbourhood U = X \ Z(f) with some G-invariant f ∈ A_{D}(X) such that the D′ ∈ Λ admitting a G-invariant f_{D′} ∈ A_{D′}(U) which is invertible in A(U) form a subgroup of finite index in Λ,

ii) stable, if x is semistable, its orbit G·x is of maximal dimension and G·x

is closed in the set of semistable points of X, iii) properly stable, if x is semistable, its isotropy group G_{x} is finite and G·x

is closed in the set of semistable points of X.

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_{0}^{s}(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_{0}^{s}(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

X̂ := Spec(A),  q: X̂ → X,  Û := q^{−1}(U).

Recall from Section 1 that the map q: X̂ → X is a geometric quotient for the action of H := Spec(𝕂[Λ]) on X̂ induced by the Λ-grading of A. Moreover, Û is a quasiaffine variety and carries a regular G-action making q: Û → 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 \overline{U} of Û such that the functions f_{i} ∈ O(Û) extend regularly to \overline{U} and \overline{U}_{f_{i}} = Û_{f_{i}} holds for each i = 1, …, r. Consider the good quotient

\overline{p}: \overline{U} → \overline{U}//G := Spec(O(\overline{U}))^{G}.

Then the quotient variety \overline{U}//G inherits a regular action of H such that the map \overline{p}: \overline{U} → \overline{U}//G becomes H-equivariant. In this setting, the set \overline{U} \ q^{−1}(X^{ss}(Λ)) takes over the role of the classical nullcone:

Proposition 2.3. Let V_{0} := \overline{U}//G \ \overline{p}(\overline{U} \ Û), and let V_{1} ⊂ \overline{U}//G be the union of all H-orbits with finite isotropy.

i) One always has q^{−1}(X^{ss}(Λ)) ⊂ \overline{p}^{−1}(V_{0} ∩ V_{1}).

ii) If U = X, then q^{−1}(X^{ss}(Λ)) = \overline{p}^{−1}(V_{0} ∩ V_{1}).

The main point in the proof is to express Condition 2.1 i) in terms of the action of the torus H on the affine variety \overline{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.

Proof of Proposition 2.3. Let W := X^{ss}(Λ) and Ŵ := q^{−1}(W). We begin with the inclusion “⊂” of assertions i) and ii). First note that Ŵ is \overline{p}-saturated, because this holds for each \overline{U}_{f_{i}} and, according to Remark 1.6 i), Ŵ is covered by these subsets. In particular, it follows \overline{p}(Ŵ) ⊂ V_{0}.

To verify \overline{p}(Ŵ) ⊂ V_{1}, let z ∈ Ŵ. Take one of the f_{i} with z ∈ \overline{U}_{f_{i}}. As it is G-invariant, f_{i} descends to an H-homogeneous function h ∈ O(\overline{U}//G). By the properties of f_{i}, the function h satisfies the condition of Lemma 2.4 for the point \overline{p}(z). Hence H_{\overline{p}(z)} is finite, which means \overline{p}(z) ∈ V_{1}. We come to the inclusion “⊃” of assertion ii). Let y ∈ V_{0} ∩ V_{1}. Lemma 2.4 provides an h ∈ O(\overline{X}//G), homogeneous with respect to some χ^{D} ∈ Char(H), such that y ∈ V := (\overline{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 V ⊂ V_{0} ∩ V_{1}.

Now, consider a point z ∈ \overline{p}^{−1}(y). Since y ∈ V_{0}, we have z ∈ X̂. We have to show that q(z) is semistable. For this, consider the G-invariant homogeneous section f := \overline{p}^{∗}(h)|X̂ 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.

Corollary 2.5. Let Λ ⊂ CDiv(X) be an ample G-linearized group.

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 X̂.

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 X̂.

3. 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].

Theorem 3.1. Let Λ ⊂ CDiv(X) be a G-linearized subgroup. Then there exists a good quotient p: X^{ss}(Λ) → X^{ss}(Λ)//G and the quotient space X^{ss}(Λ)//G is a divisorial prevariety.

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 [1, Proposition 1.5]:

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. Each h_{i} := f^{m}/g_{i} ∈ O(U_{i}) is 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 X̂ := Spec(A) and the map q: X̂ → X. Recall that the latter is a geometric quotient for the action of H := Spec(𝕂[Λ]) on X̂. Set for short W := X^{ss}(Λ). Surely, Λ is ample on W.

Proposition 1.9 yields that Ŵ := q^{−1}(W) is a quasiaffine variety. Moreover, Ŵ carries a G-action that commutes with the action of H and makes q: Ŵ → 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 h_{i} := f_{i}|_{W}.

Choose a (G × H)-equivariant affine closure \overline{W} of Ŵ such that the above h_{i} ∈ O(Ŵ) extend regularly to \overline{W} and satisfy \overline{W}_{h_{i}} = Ŵ_{h_{i}}. The set Ŵ is saturated with respect to the good quotient \overline{p}: \overline{W} → \overline{W} //G because this holds for the sets \overline{W}_{h_{i}}. Consequently, restricting \overline{p} to Ŵ yields a good quotient p̂: Ŵ → Ŵ//G. Moreover, Proposition 2.3 i) tells us that H acts with at most finite isotropy groups on Ŵ//G. Thus, there is a geometric quotient Ŵ//G → (Ŵ//G)/H. By Lemma 3.3, the quotient space is a divisorial prevariety. Since good quotients are categorical, we obtain a commutative diagram

\xymatrix{Ŵ \ar[r]^{p̂} \ar[d]_{/H} & Ŵ//G \ar[d]^{/H} \\ W \ar[r] & (Ŵ//G)/H}

Now it is straightforward to check that the induced map W → (Ŵ//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:

A(X)_{(f_{i})}^{G} ⊗ A(X)_{(f_{j})}^{G} → A(X)_{(f_{i}f_{j})}^{G}.

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_{i}^{n_{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 X := 𝕂^{2} \ {(0, 0)} via

t·(z_{1}, z_{2}) := (tz_{1}, t^{−1}z_{2}).

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

X^{ss}(Λ) = X.

To verify this claim, let A denote the graded O_{X}-algebra associated to Λ, and consider the T -invariant sections

f_{1} := 1 ∈ A_{D}(X), f_{2} := z_{1}z_{2} ∈ A_{−D}(X).

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:

1 ∈ A_{D}(X \ Z(f_{1})), \frac{1}{z_{1}z_{2}} ∈ A_{D}(X \ Z(f_{2})).

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 X := ℙ_{2} via

t·[z_{0}, z_{1}, z_{2}] := [z_{0}, tz_{1}, t^{−1}z_{2}].

Consider the group Λ ⊂ CDiv(X) generated by the divisors D_{1} := E_{0} + E_{1} and D_{2} := E_{0} + E_{2}, 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

X^{ss}(Λ) = X \ {[1, 0, 0], [0, 1, 0], [0, 0, 1]}.

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

f_{1} := 1 ∈ A_{D_{1}}(X),  f_{2} := 1 ∈ A_{D_{2}}(X),  f_{3} := \frac{z_{1}z_{2}}{z_{0}^{2}} ∈ A_{D_{1}+D_{2}}(X).

For the respective zero sets of these sections we have

Z(f_{1}) = V (X; z_{0}z_{1}),  Z(f_{2}) = V (X; z_{0}z_{2}),  Z(f_{3}) = V (X; z_{1}z_{2}).

So, the set U is indeed the union of the affine sets X \ Z(f_{i}). Moreover, we have invertible sections

1 ∈ A_{D_{1}}(X \ Z(f_{1})), \frac{z_{0}^{2}}{z_{1}z_{2}} ∈ A_{D_{2}}(X \ Z(f_{1})),

1 ∈ A_{D_{2}}(X \ Z(f_{2})), \frac{z_{0}^{2}}{z_{1}z_{2}} ∈ A_{D_{1}}(X \ Z(f_{2})),

\frac{z_{1}z_{2}}{z_{0}^{2}} ∈ A_{2D_{1}}(X \ Z(f_{3})), \frac{z_{1}z_{2}}{z_{0}^{2}} ∈ A_{2D_{2}}(X \ Z(f_{3})).

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 T - orbits 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.

4. 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:

Theorem 4.1. If U//G is divisorial, then there exists a G-linearized group Λ ⊂ CDiv(X) such that U is contained in X^{ss}(Λ) 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 E_{1}, E_{2} of E with E_{2} = h·E_{1} for some h ∈ H their multiplicities in E coincide.

Lemma 4.2. Let Λ ⊂ CDiv(Y) be a group consisting of H-tame divisors. Then Λ admits a canonical H-linearization, namely

A_{E}(U) → A_{E}(h·U), (h·f)(x) := f(h^{−1} · x).

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

ord_{E_{0}}(h·f) = ord_{h^{−1}·E_{0}}(f).

We still have to show that for every H-invariant open set V ⊂ Y, the representation of H on A_{E}(V) is regular. Consider the maximal separated subsets V_{1}, …, V_{r} of V \ Supp(E), see [3, Theorem I]. Their intersection V′ is H-invariant, 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 V_{i} = Y \ Supp(E_{i}) is covered by finitely many open affine subsets V_{ij} ⊂ V_{i} with the following properties:

i) V_{ij} = Y \ Z(h_{ij}) with some h_{ij} ∈ B_{n_{i}E_{i}}(Y), where n_{i} ∈ ℕ, ii) for each k = 1, …, r there exists an h_{ijk} ∈ B_{E_{k}}(V_{ij}) without zeroes in

V_{ij}.

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 h_{k} := 1/h′_{k} is a section of B_{E_{k}}(V) without zeroes in V. By suitably shrinking V, we achieve V = X \ Z(h) with some h ∈ B_{n_{i}E_{i}}(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

D′_{i} := p^{∗}(E_{i}) ∈ CDiv(U).

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:

i) D_{i}|_{U} = m_{i}D′_{i} holds with some m_{i} ∈ ℕ and we have X \ Supp(D_{i}) = U_{i}, ii) for some l_{i} ∈ ℕ, every f_{ij} := p^{∗}(h_{ij}^{l_{i}}) extends to a global section of O_{X}(D_{i})

and satisfies X \ Z(f_{ij}) = p^{−1}(V_{ij}).

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

f_{ijk} := p^{∗}(h_{ijk}^{m_{i}}) ∈ A_{D_{i}}(U_{ij}),

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.

Semicontinuity of the fibre dimension of p and dim(X \ U) < dim(T) imply U = X^{ss}(Λ).

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}:

X = V (ℙ_{5}; z_{0}z_{5} − z_{1}z_{4} + z_{2}z_{3}).

This allows us to define a regular action of the one dimensional torus T = ℂ^{∗} on X in terms of coordinates:

t·[z_{0}, z_{1}, z_{2}, z_{3}, z_{4}, z_{5}] := [tz_{0}, t^{2}z_{1}, t^{3}z_{2}, t^{3}z_{3}, t^{4}z_{4}, t^{5}z_{5}].

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):

f_{01} := \frac{z_{0}^{2}z_{4}}{z_{1}^{3}} ∈ A_{4D_{1}−D_{2}}(X), h_{01} := 1 ∈ A_{D_{1}}(X \ Z(f_{01})),

f_{02} := \frac{z_{0}z_{2}}{z_{1}^{2}} ∈ A_{2D_{1}}(X), h_{02} := \frac{z_{2}^{3}}{z_{0}z_{4}^{2}} ∈ A_{2D_{2}}(X \ Z(f_{02})),

f_{03} := \frac{z_{0}z_{3}}{z_{1}^{2}} ∈ A_{2D_{1}}(X), h_{03} := \frac{z_{3}^{3}}{z_{0}z_{4}^{2}} ∈ A_{2D_{2}}(X \ Z(f_{03})),

f_{04} := \frac{z_{0}^{2}z_{4}}{z_{1}^{3}} ∈ A_{3D_{1}}(X), h_{04} := 1 ∈ A_{D_{2}}(X \ Z(f_{04})),

f_{05} := \frac{z_{0}^{3}z_{5}}{z_{1}^{4}} ∈ A_{4D_{1}}(X), h_{05} := \frac{z_{0}z_{5}^{3}}{z_{4}^{4}} ∈ A_{4D_{2}}(X \ Z(f_{05})),

f_{12} := \frac{z_{2}^{2}}{z_{1}z_{4}} ∈ A_{2D_{1}+D_{2}}(X), h_{12} := 1 ∈ A_{D_{1}}(X \ Z(f_{12})),

f_{13} := \frac{z_{3}^{2}}{z_{1}z_{4}} ∈ A_{2D_{1}+D_{2}}(X), h_{13} := 1 ∈ A_{D_{1}}(X \ Z(f_{13})),

f_{14} := 1 ∈ A_{D_{1}+D_{2}}(X), h_{14} := 1 ∈ A_{D_{1}}(X \ Z(f_{14})),

f_{15} := \frac{z_{1}z_{5}^{2}}{z_{4}^{3}} ∈ A_{3D_{2}}(X), h_{15} := 1 ∈ A_{D_{1}}(X \ Z(f_{15})),

f_{24} := \frac{z_{2}^{2}}{z_{1}z_{4}} ∈ A_{D_{1}+2D_{2}}(X), h_{24} := 1 ∈ A_{D_{2}}(X \ Z(f_{24})),

f_{25} := \frac{z_{2}z_{5}}{z_{4}^{2}} ∈ A_{2D_{2}}(X), h_{25} := \frac{z_{2}^{3}}{z_{1}^{2}z_{5}} ∈ A_{2D_{1}}(X \ Z(f_{25})),

f_{34} := \frac{z_{3}^{2}}{z_{1}z_{4}} ∈ A_{D_{1}+2D_{2}}(X), h_{34} := 1 ∈ A_{D_{2}}(X \ Z(f_{34})),

f_{35} := \frac{z_{3}z_{5}}{z_{4}^{2}} ∈ A_{2D_{2}}(X), h_{35} := \frac{z_{3}^{3}}{z_{1}^{2}z_{5}} ∈ A_{2D_{1}}(X \ Z(f_{35})),

f_{45} := \frac{z_{1}z_{5}^{2}}{z_{4}^{3}} ∈ A_{4D_{2}−D_{1}}(X), h_{45} := 1 ∈ A_{D_{2}}(X \ Z(f_{45})).

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).

5. 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.

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 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.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 χ̃ ∈ Char(N(T)), the restriction χ := χ̃|_{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]. 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 we obtain a commutative diagram

\xymatrix{X \ar[rr]^{//N(T)} \ar[dr]_{//R} &  & X//N(T) \\  & X′ \ar[ur]}

Consider the induced action of the connected semisimple group G′ := G/R on X′. The image T′ of T under the projection G → G′ is a maximal torus of G′. Moreover, N(T′) is the image of N(T) under G → G′. Thus, the upwards arrow of the above diagram is a good quotient for the action of N(T′) on X′. To proceed, we only have to derive from the existence of a good quotient X′ → X′//N(T′) that there is a good quotient X′ → X′//G′ with a divisorial prevariety X′//G′. In other words, we may assume from the beginning that the group G is semisimple. Let p: X → X//N(T) denote the good quotient. Using Lemmas 4.2 and 4.3 we can construct a canonically N(T)-linearized ample group Λ ⊂ X consisting of N(T)-tame divisors such that we have

X^{ss}(Λ, N(T)) = X.

Note that this equality also holds for any subgroup Λ′ ⊂ Λ of finite index in Λ. We construct now such a subgroup Λ′ ⊂ Λ for which the canonical N(T)-linearization of Λ′ extends to a strong G-linearization. The first step is to realize X as an open G-invariant subset of a certain G-prevariety Y with O(Y) = 𝕂. Consider the maximal separated open subsets X_{1}, …, X_{m} ⊂ X, see [3, Theorem I]. Since G is connected, it leaves these sets invariant. By Sumihiro’s Equivariant Completion Theorem [23, Theorem 3], we find G-equivariant open embeddings X_{i} → Z_{i} into complete G-varieties Z_{i}. Applying equivariant normalization, we achieve that each Z_{i} is normal. Let Y_{i} denote the union of X_{i} with the set of regular points of Z_{i}. Note that O(Y_{i}) = 𝕂. Define Y to be the G-equivariant gluing of the varieties Y_{i} along the invariant open subsets X_{i} ⊂ Y_{i}. Then we have O(Y) = 𝕂. Moreover, all points of Y \ X are regular points of Y. By closing components, every Cartier divisor D ∈ Λ extends to a Cartier divisor on Y. Let Γ ⊂ CDiv(Y) denote the (free) group of Cartier divisors generated by these extensions. Lemma 4.2 ensures that the canonical N(T)-linearization of Λ extends to a canonical N(T)-linearization of the group Γ. By [23, Corollary 2] and Proposition 1.3, this linearization is even a strong one. We claim that some subgroup Γ′ ⊂ Γ of finite index admits a strong G-lineariza-tion. Let B be the graded O_{Y} -algebra associated to Γ. For each homogeneous component B_{E,i} := B_{E}|_{Y_{i}}, some power B_{nE,i} admits a G-linearization as in [16, Proposition 2.4]. Since G is semisimple, these linearizations are unique, see [19, Proposition 1.4]. Thus they define G-sheaf structures on the O_{Y_{i}}-algebras

B_{i} := ⨁_{E∈Γ_{i}} B_{E,i},

for suitable subgroups Γ_{i} ⊂ Γ of finite index. Again by uniqueness of strong G-linearizations, we can patch the above G-sheaf structures together to the desired strong G-linearization on the intersection Γ′ ⊂ Γ of the subgroups Γ_{i} ⊂ Γ, and our claim is proved. Now, since the character group of N(T) is finite, Proposition 1.5 tells us that on some subgroup Γ′′ ⊂ Γ′ of finite index, the canonical N(T)-linearization and the one induced by the G-linearization coincide. Thus restricting Γ′′ to X provides the desired subgroup Λ′ ⊂ Λ of finite index. We replace Λ with Λ′. In order to obtain a quotient of X by G, we want to apply Theorem 3.1. So we have to show that X^{ss}(Λ, G) equals X. For this, let A be the graded O_{X}- algebra associated to Λ, and set X̂ := Spec(A). Moreover, let q: X̂ → X be the canonical map and H := Spec(𝕂[Λ]) the torus acting on X̂. Note that

X^{ss}(Λ, T) = X^{ss}(Λ, N(T)) = X.

Choose G-invariant homogeneous f_{1}, …, f_{r} ∈ A(X) and T -invariant homogeneous h_{1}, …, h_{s} ∈ A(X) such that the complements X \ Z(f_{i}) and X \ Z(h_{i}) satisfy the condition of Definition 2.1 i) and

X^{ss}(Λ, G) = (X \ Z(f_{1})) ∪ … ∪ (X \ Z(f_{r})),
X^{ss}(Λ, T) = (X \ Z(h_{1})) ∪ … ∪ (X \ Z(h_{r})).

Since Λ is ample, Proposition 1.9 yields a (G × H)-equivariant affine closure \overline{X} of X̂ such that the f_{i} and the h_{j} extend to regular functions on \overline{X} satisfying \overline{X}_{f_{i}} = X̂_{f_{i}} and \overline{X}_{h_{j}} = X̂_{h_{j}}. Moreover, we obtain a commutative diagram of H-equivariant maps:

\xymatrix{\overline{X} \ar[rr]^{^{\overline{p}^{G}}}_{//G} \ar[dr]_{_{\overline{p}_{T}}}^{//T} &  & \overline{X}//G \\  & \overline{X}//T \ar@{-->}[ur]}

Now, let x ∈ X, and assume that x is not semistable with respect to G. Choose z ∈ q^{−1}(x), and let y := \overline{p}_{G}(z). By Proposition 2.3 ii), the assumption x ∉ X^{ss}(Λ, G) amounts to y ∈ \overline{p}_{G}(\overline{X} \X̂) or to an isotropy group H_{y} of positive dimension. First suppose that we have y ∈ \overline{p}_{G}(\overline{X} \ X̂). Let G·z′ be the closed orbit in \overline{p}_{G}^{−1}(y). Then G·z′ is contained in \overline{X} \ X̂. Moreover, the Hilbert-Mumford-Birkes Lemma [10], provides a maximal torus T′ ⊂ G such that the closure of T′·z intersects G·z′.

Let g ∈ G with gT′g^{−1} = T. Then the closure of T·g·z contains a point z′′ ∈ G·z′. Surely, \overline{p}_{T} (g·z) equals \overline{p}_{T} (z′′). Thus, since z′′ ∈ \overline{X} \ X̂, 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 ∈ \overline{p}_{G}(\overline{X} \X̂) is excluded, the isotropy group H_{y} is of positive dimension, and the whole fibre \overline{p}_{G}^{−1}(y) is contained in X̂. Let H_{0} ⊂ H_{y} be the connected component of the neutral element. Then H_{0} acts freely on the fibre \overline{p}_{G}^{−1}(y), and the closed orbit G·z′ ⊂ \overline{p}_{G}^{−1}(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.

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Jürgen Hausen

Fachbereich Mathematik und Statistik Universität Konstanz 78457 Konstanz Germany hausenj@fmi.uni-konstanz.de