Invariants and Separating Morphisms for Algebraic Group Actions

The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient $X/\!/\!G$ given by the possibly not finitely generated ring of invariants is"almost"an algebraic variety, and that the quotient morphism $\pi\colon X \to X/\!/\! G$ has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective. These general results are then refined for actions of the additive group $\mathbb{G}_a$, where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separating variety, which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts' famous example of a an action of $\mathbb{G}_a$ on 7-dimensional affine space with a non-finitely generated ring of invariants.


Introduction
In all classification problems invariants play an important rôle. They let one distinguish nonequivalent objects, characterize specific elements, or detect certain properties. For instance, the genus of a complex smooth projective curve C determines the topology of the compact surface C, and the discriminant of a polynomial tells us whether it has multiple roots. But there are many other examples of important invariants, like the Alexander-polynomial of a knot or the Dedekind ζ -function of a number field.
In the algebraic setting where we work over an algebraically closed field k, we can often reduce a classification problem to the following general situation. There is an algebraic variety X representing the objects, and an algebraic group G acting on X such that two objects x, y ∈ X are equivalent if and only if they belong to the same orbit under G. In this case the classification problem amounts to describing the orbit space X/G. Clearly, X/G inherits some properties from X : it has a topology and the (continuous) functions on X/G correspond to the (continuous) G-invariant functions on X . Of course, we would like to see X/G again as an algebraic variety, but this cannot work in general, because X usually contains nonclosed orbits, and so X/G contains nonclosed points.
If X is an affine variety with coordinate ring O(X ), we can look at the subalgebra O(X ) G ⊂ O(X ) of G-invariant functions and consider the morphism π X : X → X/ /G := Spec O(X ) G induced by the inclusion. It is a categorical quotient in the category of affine k-schemes, and has the usual universal property: Every G-invariant morphism X → Y factors uniquely through π X . In some sense this is the best schematic approximation to the orbit space. We will say that X/ /G is the quotient scheme and π X : X → X/ /G the quotient morphism or shortly the quotient.
If G is reductive, then O(X ) G is finitely generated and so X/ /G is an affine variety. Moreover, π X has some nice properties ([19, chap. 1.2 Theorem 1.1]): • π X is G-closed: If Z ⊂ X is G-stable and closed, then π X (Z ) is closed.
• π X is G-separating: If Z , Z ′ ⊂ X are disjoint G-stable closed subsets, then π X (Z ) ∩ π X (Z ′ ) = ∅. In particular, π X is surjective and every fiber contains a unique closed orbit. Thus the variety X/ /G classifies the closed orbits in X . In good situations, the general orbits are closed, and so, at least generically, X/ /G is the orbit space.
If G is not reductive, then all this fails to be true. In particular, the invariant ring might not be finitely generated and so the quotient X/ /G is not an algebraic variety, and the quotient morphism π X is usually not surjective. The fact that X/ /G is not of finite type is considered to be the main difficulty in handling non-reductive groups. We think that the non-surjectivity of π X is even a more serious problem.
One of the aims of this paper is to show that the quotient X/ /G as a k-scheme is "almost algebraic" in the following sense. An open subset U of a k-scheme is called an algebraic variety or shortly algebraic if U , as a reduced scheme, is separated and of finite type. (The separatedness is generally not an issue here, because we are working with affine schemes.) Then we show that X/ /G contains large open algebraic subsets and that it shares many properties with algebraic varieties. This is explained in Sects. 2 and 4 which are inspired by Winkelmann's work [25]. For example, if the base field is uncountable, then X/ /G is a Jacobson scheme which implies that the Zariski topology on X/ /G is determined by the Zariski topology on the k-rational points of X/ /G.
To have an idea of our approach and our results let us give a geometric interpretation of Roberts famous example of an action of the additive group G a = (k, +) on A 7 with a non-finitely generated ring of invariants. The details are given in the last Sect. 9. Let π : A 7 → A 7 / /G a be the quotient. (a) The fixed point set F := (A 7 ) G a ≃ A 4 is mapped under π to a single point π(0) ∈ A 7 / /G a ; (b) The complement A 7 bd := A 7 \F is a principal G a -bundle over its image π(A 7 bd ) ⊂ X/ /G a which is an open algebraic subset and contains every open algebraic subset U of A 7 / /G a ; (c) The image of π is π(A 7 ) = π(A 7 bd ) ∪ {π(0)}. (d) The complement (A 7 / /G a )\π(A 7 bd ) is isomorphic to A 3 . An important new feature is the concept of a separating morphism ϕ : X → Y where Y is an algebraic variety (cf. [3, section 2.3]). This means that ϕ is G-invariant and separates the same orbits as π X does. Such morphisms always exist even when the invariants are not finitely generated, but finding a "nice" separating morphism is usually a difficult task. For Roberts example we get the following.
(e) There exists a separating morphism ϕ : A 7 → A 9 such that Y := ϕ(A 7 ) is normal of dimension 6. (f) The induced mapφ : A 7 / /G a → Y is injective. It defines a homeomorphism π(A 7 ) → ϕ(A 7 ) and an isomorphism π(A 7 bd ) . Another important concept is the separating variety which measures how much the invariants separate the orbits. It is defined as the reduced fiber product S X := X × X/ /G X and contains the closure of the graph X := {(gx, x) | g ∈ G, x ∈ X }. If a general fiber of the quotient map is an orbit and if G is connected, then X is an irreducible component of the separating variety. But even in nice situations, the separating variety may have additional components. In general, the meaning of these other components is not yet well understood, except for some special cases (see below). For Roberts' example we find the following.
(h) The separating variety has two irreducible components: The most complete results are obtained for actions of the additive group G a , in particular for representations of G a (Sects. 5-7). This part of our work was inspired by certain calculations done by Elmer and Kohls in [8]. An important tool here is the geometric interpretation of the zero set P X of the plinth ideal (Definition 5.2). If X is factorial, then X \P X is equal to the open set X bd where X is locally a G a -bundle. In Sect. 8 we generalize some of the results for representations of G a to G a -actions induced by actions of SL 2 .
To prepare the reader for the difficulties in working with non-finitely generated algebras we describe an easy example in Sect. 3.

Invariants
Our base field k is algebraically closed. In the second part, starting with Sect. 5, we study G a -actions and will assume that char k = 0. Since we have to deal with non-finitely generated rings of invariants, we will work in the category of reduced k-schemes Z . However, from the geometric point of view we are mainly interested in the k-rational points of Z which will denote by Z (k). In this setting, a variety Z is a reduced separated k-scheme of finite type, and in this case we will often confuse the scheme Z with its k-rational points Z (k).
Throughout this paper, we let X be a normal affine variety and G an algebraic group acting on X . We denote by O(X ) the k-algebra of regular functions on X and by O(X ) G ⊂ O(X ) the subalgebra of G-invariant functions. The quotient is defined to be the affine k-scheme If the base field k is uncountable, a famous result of Krull's implies that X/ /G is a Jacobson scheme, i.e., O(X ) G is a Jacobson ring [13]. This means that every radical ideal of O(X ) G is the intersection of maximal ideals. Moreover, every closed point of X is k-rational in this case, since O(X ) G is contained in a finitely generated k-algebra. It follows that the Zariskitopology on X/ /G is completely determined by the Zariski-topology on the k-rational points (X/ /G)(k). This allows to work with k-rational points which are the only interesting objects from a geometric point of view, as mentioned above.
Remark 2.1 If the k-algebra R is not a Jacobson ring, then there is a prime ideal p ⊂ R which is not the intersection of the maximal ideal containing p. In geometric terms this means the following. Denote by Z ⊂ Spec R the closed subscheme defined by p, and let Z cl ⊂ Z be the subset of closed points. Then the closure Z cl in Spec R is strictly contained in Z .

Quotient morphism
Although O(X ) G might not be finitely generated, hence X/ /G is not of finite type, we will see that the quotient X/ /G contains large open subschemes which are varieties. For this we need the following result due to Derksen and Kemper [4, Propositions 2.7 and 2.9].

Proposition 2.2 Let R be a k-algebra. Define
Then f R is a radical ideal of R. If R is contained in a finitely generated k-domain, then f R ̸ = (0).
The ideal f R will be called the finite generation ideal. Remark 2.3 The open subset Spec R\V(f R ) ⊂ Spec R is the union of all open subsets U ⊂ Spec R which are algebraic. In fact, each such U is a finite union of open affine varieties U i , and each U i is a finite union of some (Spec R) f j . We will denote the complement Spec R\V(f R ) by (Spec R) alg : Note that (Spec R) alg is itself a variety if and only if f R is the radical of a finitely generated ideal. On the other hand, (Spec R) alg is always Jacobson and its closed points coincide with its k-rational points.

Definition 2.4
Let Z = Spec R be an affine k-scheme. If A ⊂ Z is a closed subset we define I (A) ⊂ R to be the (radical) ideal of functions vanishing on A.

(d) If
A ⊂ Z is closed, then codim Z A := min{ht p | p ⊃ I (A), p prime} where ht p is the height of the prime ideal p.
As an example, we will see later in Theorem 4.3(a) that the quotient X/ /G is always finite dimensional, and that dim X/ /G = tdeg k O(X ) G .

Algebraic varieties
Assume that Z = Spec R is a variety. Then Z = i Z i is a finite union of irreducible closed subsets, and dim Z = max i {dim Z i }. Moreover, if Z is irreducible, then dim Z = tdeg k R, and for every irreducible closed subset Finally, if ϕ : Z → Y is a morphism where Y is an arbitrary reduced k-scheme, and if A ⊂ Z is a closed subscheme, then ϕ(A(k)) is dense in ϕ(A) ⊂ Y . As mentioned before, this last statement holds more generally if Z is a Jacobson scheme.

A first example
Let us discuss an interesting example. While it does not quite fit in our setting-it does not arise from a quotient of an algebraic group action on a normal affine variety-it has a similar behavior.
Consider the graded subring R := k[x, x y, x y 2 , x y 3 , . . .] ⊂ k[x, y] generated by the monomials x y k , k = 0, 1, . . ., and set Z := Spec R. (c) π : A 2 → Z is surjective and induces an isomorphism Finally, we consider the affine morphism ϕ : A 2 → A 2 given by (x, y) → (x, x y). (This morphism plays the role of a separating morphism.) (e) ϕ factors through π andφ is injective. Hence ϕ separates the same points of A 2 as π does.
The proofs are not difficult and are left to the reader. They are based on the following lemma.

Separation
The so-called separation property will play an important role in this paper.
is an affine variety is a separating morphism if it satisfies the following Separation Property:

Remark 4.2
If char k = 0, then the separation property (SP) implies that ϕ * induces an isomorphism k(ϕ(X )) ∼ − → k(X/ /G). If char k > 0, we say that ϕ is strongly separating if ϕ is separating and induces an isomorphism k(ϕ(X )) It is shown in [3, Theorem 2.3.15] that separating morphisms always exist. In more algebraic terms this means that one can find a finitely generated separating subalgebra R ⊂ O(X ) G , i.e., a subalgebra which separates the same k-rational points of X as the invariant functions. We can always add invariant functions to R, and thus assume that R is normal and that Q(R) = k(X/ /G), if necessary. Thus, a strongly separating morphism ϕ : X → Y with Y normal always exists. A basic problem is to find a separating algebra with a small number of generators.

Main results
A G-invariant morphism ϕ : X → Y where Y is an affine variety always factors through the quotient morphism π : X → X/ /G: Then ϕ is separating if and only ifφ is injective on the image π(X (k)) ⊂ (X/ /G)(k) of the k-rational points. In the paper [25] Winkelmann studies this general set-up and proves a number of fundamental results, e.g. that every such invariant ring O(X ) G is the ring of global regular functions on a quasi-affine variety and vice versa. Some of his results are contained and extended in the following theorem, where we take a geometric point of view. From that point of view we are mainly interested in the images π(X ) ⊂ X/ /G and ϕ(X ) ⊂ Y and how they are related to (X/ /G) alg = X/ /G\V(f X/ /G ) where f X/ /G ⊂ O(X ) G denotes the finite generation ideal (Proposition 2.2, Remark 2.3).

Theorem 4.3
Let X be a normal affine variety with an action of an algebraic group G and denote by π : X → X/ /G the quotient morphism. Let ϕ : X → Y be a dominant separating morphism where Y is a normal affine variety.
(a) If A ⊂ X is an irreducible closed subset, then dim π(A) = dim ϕ(A) and Now assume that ϕ is strongly separating, and let C Y := Y \ϕ(X ) be the complement of the image of ϕ.
Let us draw some diagrams. Suppose ϕ is strongly separating. The statements (b) and (f) give and from (e) we have From the statements (b) and (g) we get

Corollary 4.6 If V is a rational representation of G and if
Proof This is clear from the previous corollary, because O(V ) G does not contain nonconstant invertible functions. ⊓

Remark 4.7
In the case where G is reductive, this last corollary is an easy consequence of Igusa's Criterion [10, Lemma 4].
We say that an affine k-scheme Z = Spec R is a cone with apex z 0 , if R = i≥0 R i is a positively graded ring with R 0 = k and z 0 is the homogeneous maximal ideal. Geometrically this means that Z admits an action of the multiplicative group G m := k * with a single closed orbit, namely the fixed point z 0 . An affine variety X is called a G-cone if X is a cone and the G-action commutes with the G m -action. In particular, the apex x 0 is a fixed point for G. In this case (X/ /G, π(x 0 )) is a cone, and the finite generation ideal f X/ /G is homogeneous.
Proof The complement of (X/ /G) alg in X/ /G is a closed cone, hence empty, because it does not contain the apex. Thus O(X ) G is finitely generated. Sinceφ :

Proof of Theorem 4.3
The proof needs some preparation.

is open, algebraic and dense in Z .
Proof By first inverting some f ∈ f R we can assume that R is finitely generated. In this case the result is well known, cf.  (a) R = p∈P R p where P is the set of the primes of R of height 1; (b) R p is a discrete valuation ring for all p ∈ P; (c) For any nonzero r ∈ R the set {p ∈ P | p ∋ r } is finite.

Lemma 4.12 For any r
Proof We can assume that r is neither zero nor invertible. Then r O(X ) G = finite p (n p ) is a finite intersection of symbolic powers ([16, §12, page 88, Corollary to Theorem 12.3]). Hence V X/ /G (r ) = finite S i where S i are irreducible closed subschemes of codimension 1. Now the claim follows from the previous lemma. ⊓ We will also need the following result; the proof is easy and left to the reader.

Lemma 4.13 Let Y be an irreducible variety, C ⊂ Y an irreducible closed subset of codimension d and U ⊂ Y a nonempty open set. Then there is a chain
is a bijective morphism of varieties, we see that, for j < d, If not, using again Lemma 4.9, we can find a subset U ⊂ π(A) which is open and dense in π(A) and such that The same argument as above shows that, for irreducible closed subsets A, B ⊂ X with π(A) π(B), we have ϕ(A) ϕ(B). It follows that the map π(X ) → ϕ(X ) is injective, hence bijective, and open, hence a homeomorphism. (c) For p ∈ X/ /G we have p ∈ C X = (X/ /G)\π(X ) if and only if π(V X (p)) V(p) where V(p) denotes the zero set in X/ /G. Assume now that codim X/ /G C X = 1. This means that C X contains an irreducible closed subscheme S of codimension 1 corresponding to a prime ideal p ∈ C X of height 1. It follows that π(π −1 (S)) S, contradicting Lemma 4.11.
(d) Let S ⊂ X/ /G be an irreducible hypersurface and let p ⊂ R := O(X ) G be the corresponding prime ideal of height 1. Then, by Lemma 4.11 and (b), H :=φ(S) is an irreducible hypersurface, and so the corresponding prime ideal p ′ : From now on we assume that char k = 0. In this and the following sections we focus on G a -varieties, i.e., varieties with an action of the additive group G a ≃ (k, In this case, Y can be identified with the orbit space X/G a , and the quotient morphism π : X → X/G a admits a section. If X is affine, then X/G a = Spec O(X ) G a , and this is an algebraic variety.
The G a -variety X is called a principal G a -bundle (for short, a G a -bundle) if there is a G ainvariant morphism π : X → Z and an open covering Z = i U i such that p −1 (U i ) → U i is a trivial G a -bundle for all i. In this case, Z can be identified with the orbit space X/G a and the morphism π has the usual universal properties. Again, if X is affine, then X/G a = Spec O(X ) G a , and this is an algebraic variety.

Local slices
Now let X be an affine G a -variety. The G a -action defines a locally nilpotent vector field D ∈ Vec(X ) := Der k (O(X )) which determines the G a -action. Its kernel coincides with the ring of invariants: The corresponding (reduced) closed subscheme P X/ /G ⊂ X/ /G is called the plinth scheme of X/ /G whereas the zeros set P X := V X (p X/ /G a ) ⊂ X is called the plinth variety of X . By definition, we have P X = π −1 (P X/ /G ), and p X/ /G a ⊆ f X/ /G a .
The next result shows that outside the plinth variety the quotient morphism is a principal G a -bundle. Proposition 5. 2 We have π(X \P X ) = X/ /G a \P X/ /G a and this is an open algebraic variety of X/ /G a . Moreover, the morphism π : X \P X → X/ /G a \P X/ /G a is a principal G a -bundle.
Proof If s = D f and Ds = 0, then π(X s ) = (X/ /G a ) s , and this is an open subset of X/ /G a which is an affine variety. Since we can cover X \ P X with finitely many X s j we see that π(X\P X ) = s∈p X/ /Ga (X/ /G) s = X/ /G\P X/ /G is also covered by finitely many open affine varieties (X/ /G) s j , hence is a variety. It remains to see that π separates the G a -orbits on X\P X . This is clear for two orbits contained in the same X s j . If O 1 ⊂ X s j and O 2 ⊂ X s k \X s j , then the invariant s j vanishes on O 2 , but not on O 1 . ⊓ Definition 5.3 Let X be a G a -variety. Define X bd ⊆ X to be the union of all open G a -stable subsets U which are trivial G a -bundles: If X is affine, it follows from Proposition 5.2 that X \ P X ⊆ X bd . We will see later (Example 8.4) that the inclusion may be strict. However, this cannot happen if X is factorial.

Proposition 5.4 Let X be a factorial affine G a -variety. Then
In particular, π(X bd ) ⊆ X/ /G a is an open subvariety, and X bd → π(X bd ) is a principal G a -bundle.
Proof In the definition of X bd we can assume that all U i are affine. Since X is factorial, this implies that U i = X t i for a suitable invariant t i . On the other hand, if X t is a trivial G a -bundle where t ∈ O(X ) G a , then there is an h ∈ O(X t ) such that Dh = 1. Writing h = f t −k we see that s := t k = D f , and so X s = X t is of the form above.
This defines a decomposition of V into weight spaces:

123
Since the invariants are finitely generated, the quotient V / /G a := Spec O(X ) G a is an affine variety. As usual, the nullcone is defined by N = N V := π −1 (π(0)) ⊆ V . Recall that the Weyl-group W ≃ Z/2Z of SL 2 acts on the zero weight space V 0 = V G m . The nontrivial element of W is represented by the matrix σ = 0 −1 1 0 ∈ SL 2 .
given by the SL 2 -invariants and has a factorization where π 0 is the quotient by W andπ is finite and bijective. Remark 6.2 Elmer and Kohls [8] gave an explicit construction of separating sets for indecomposable representations, which were later extended to any representation by Dufresne, Elmer, and Sezer [6].
The proof of the theorem needs some preparation.

Invariants and covariants
Let V be a representation of SL 2 . The graded coordinate ring O(V ) = d≥0 O(V ) d is a locally finite and rational SL 2 -module. A homogeneous irreducible submodule F ⊂ O(V ) d is classically called a covariant of degree d and weight r, where r is the weight of the highest weight vector f 0 of F. This means that f 0 is a homogeneous G a -invariant and that t · f 0 = t r f 0 for t ∈ G m . In particular, dim F = r + 1. Thus, we always have r ≥ 0, and r = 0 if and only if f 0 is an SL 2 -invariant. We will say that f 0 is a homogeneous G a -invariant of degree d and weight r .
Clearly, the invariants O(V ) G a are linearly spanned by the homogeneous G a -invariants of degree d and weight r where d, r ≥ 0. Moreover, the homogeneous G a -invariants of degree d and weight r > 0 linearly span the plinth ideal p V = ker D ∩ im D where D ∈ Vec(V ) is the locally nilpotent vector field corresponding to the G a -action (see Definition 5.1). This shows that the G a -invariants are generated by p V together with the SL 2 -invariants.
In the following, we denote by V [n] the irreducible SL 2 -module of highest weight n, i.e., dim V [n] = n + 1. One can take V [n] := k[x, y] n , the binary forms of degree n, with the standard linear action of SL 2 . It follows that the element σ ∈ SL 2 representing the nontrivial element of the Weyl group acts trivially on V [n] 0 if n is odd or n ≡ 0 (mod 4), and by (− id) if n ≡ 2 (mod 4).
In the proof below we will need the following classical result from invariant theory of binary forms. Choose a basis of weight vectors of V [n] such that O(V [n]) = k[x 0 , x 1 , . . . , x n ], where x i has weight n − 2i.
, where x i has weight n − 2i. Thus x i vanishes on V + if and only if 2i ≤ n, and x i vanishes on V + ⊕ V 0 if and only if 2i < n. Now let v = (a 0 , a 1 , . . . , a n ) ∈ V \(V + ⊕ V 0 ), and let a k be the first nonzero coefficient. Then the quadratic G a -invariant f k from Lemma 6.3 above gives f k (v) = α k a 2 k ̸ = 0, and since k < n/2 the G a -invariant f k has a positive weight. It remains to show that every homogeneous G a -invariant f of weight > 0 vanishes on V + ⊕ V 0 . But this is clear, because every monomial m = x d 0 0 x d 1 1 . . . x d n n of positive weight must contain an x i of positive weight, i.e., with 2i < n. Hence m vanishes on V + ⊕ V 0 . (c) The same argument shows that a homogeneous SL 2 -invariant restricted to V + ⊕ V 0 does not depend on V + . This implies that the induced morphism π| P V : P V → π(P V ) ⊆ V / /G a is given by the SL 2 -invariants and has the following factorization where π SL 2 : V → V / / SL 2 is the quotient by SL 2 . The following lemma shows that π SL 2 | V 0 induces a finite bijective morphism V 0 /W → π(P V ), as claimed. ⊓ The following general result was pointed out to us by the referee.

Lemma 6.4 Let G be a connected reductive group with maximal torus T and Weyl group W . For any affine G-variety Z the natural map Z T /W → Z / /G is finite and injective.
Proof The finiteness follows from [15, 2.1 Théorème]. Also, for any z ∈ Z T , the orbit Gz ⊂ Z is closed, since the stabilizer contains a maximal torus, and Gz ∩ X T is a unique W -orbit, because all maximal tori in G z are conjugate. ⊓ 123 7 The separating variety

Definitions
In Sect. 4, we discussed separating morphisms in the general context of a G-variety. We now introduce the separating variety S X of a G-variety X , which measures how much the invariants separate the orbits (see [11,Section 2]). Set where π : X → X/ /G is the quotient morphism. More schematically, the separating variety of X is the reduced fiber product The separating variety S X contains the closure of the graph Note that X = S X exactly when the quotient π is almost geometric, i.e., when all nonempty fibers of π are orbits. Also, if X is closed, then all orbits are closed and have the same dimension. (The first statement is clear, and the second follows since More generally, we have the following result, which is a first step to determine the closure X and to decide whether X = S X . For simplicity, we assume that G is connected which implies that X is irreducible.

Proposition 7.1 Let G be connected and X a normal affine G-variety. Assume that there is
a dense open set U ⊆ X/ /G such that ϕ −1 (u) is nonempty and contains a dense orbit for all closed points u ∈ U . Set X ′ := π −1 (U ) ⊆ X and P := X \X ′ .
(a) S X,P is closed and S X = X ∪ S X,P . In particular, X is an irreducible component of S X . (b) If π −1 (u) is a single orbit for every closed point u ∈ U , then S X = X ′ ∪ S X,P = X ∪ S X,P = X ∪ S X,P .
(c) Assume in addition that X ′ is smooth, that the G-action on X ′ is free, and that codim X P > 1. Then either X is closed, or X \ X ′ has codimension 1 in X .
is closed, and the union is disjoint. Take (x, y) ∈ S X,X ′ . Then π(x) = π(y) =: u ∈ U . By assumption, the fiber π −1 (u) contains a dense orbit, say Gz = π −1 (u). Hence, It follows that S X = S X,X ′ ∪ S X,P = X ∪ S X,P . (b) Since the fibers over U are orbits, we get S X,X ′ = X ′ = X ∩ (X ′ × X ′ ), and so S X = S X,X ′ ∪ S X,P = X ′ ∪ S X,P .
The claim follows.
By assumption, it induces an isomorphism µ 0 : G × X ′ ∼ − → X ′ , and thus, a birational morphismμ : G×X →˜ , where˜ → X is the normalization. If codim X X \ X ′ > 1, then by Igusa's criterion [10],μ is an isomorphism, and so X is closed. ⊓ Remark 7.2 The first statement of the proposition above has the following converse: If X is an irreducible component of S X , then the general fiber of π : X → X/ /G contains a dense orbit.
In order to see this, we can replace X/ /G be a dense open set and thus assume that X/ /G is affine algebraic, that π : X → X/ /G is flat, and that the fibers are irreducible of dimension n. Then every irreducible component of S X = X × X/ /G X has dimension 2 dim X − dim X/ /G = dim X + n (see [9,Corollary 9.6 in Chap. III]). On the other hand, dim X = dim X + d where d := max{dim Gx | x ∈ X }. Hence n = d and so the general fiber contains a dense orbit.

The case of G a -varieties
If X is a G a -variety, then, by Proposition 5.2, the quotient π : X \P X → π(X \P X ) is a G a -bundle. This implies the following corollary.

and X is an irreducible component of S X .
In the remaining part of this section, we determine the irreducible components of S V for a representation V of G a (cf. [7], where this is done for indecomposable representations). We have seen in Theorem 6.1(c) that the image π(P V ) ⊂ V / /G a is closed and the induced morphism π| P V : P V → π(P V ) has a factorization where π 0 is the linear projection onto W andπ is finite and bijective. If v ∈ P V = V 0 ⊕ V + , we denote by v 0 the component of v in V 0 . Define the following closed subsets of S P V : Both are irreducible and isomorphic to V 0 × (V + × V + ). Now the factorization ( * ) implies the following result.

Lemma 7.4
(a) If σ acts trivially on V 0 , then

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Proof We can assume that V SL 2 = (0). In fact, if V = W ⊕ k m , then V = W × k m and S V = S W × k m . It is easy to see that for V = V [2] we have S V = V . In all other cases, we have dim V + ≥ 2 which implies that the component C is not contained in V . On the other hand, V = V ∪ C σ by Lemma 7.6 below, and the claim follows from Lemma 7.4. ⊓ Lemma 7. 6 We The proof needs some preparation. If X is a variety and R a k-algebra, we define the R-valued points by X (R) := Mor(Spec R, X ). Denote by k[[t]] the power series ring and by k((t)) its field of fractions. We have a canonical inclusion X (k[[t]]) ⊂ X (k((t))) and a canonical map X (k[[t]]) → X (k) = X which will be denoted by x = x(t) → x(0) = x| t=0 . We will constantly use the following known fact. For completeness we include a short proof. Lemma 7.7 If ϕ : X → Y is a morphism and y ∈ ϕ(X ), then there is an Proof We first claim that there is an irreducible curve D ⊂ Y such that y ∈ D and D ∩ ϕ(X ) is open and dense in D. This is obvious if Y = k n . In general, we can assume that Y is normal and dim Y > 1. Then we choose a finite surjective morphism ψ : Y → k n and use the Going-down property of ψ to show that there is an irreducible hypersurface in H ⊂ Y which contains y and meets ϕ(X ) in a dense set. Now the claim follows by induction. (A stronger result can be found in [17, Lemma on page 56]: Any two points of an irreducible variety can be connected by an irreducible curve.) As a consequence we see that there is a smooth curve C, a point c ∈ C and a morphism ρ :

G a -actions on SL 2 -varieties
In this section, we generalize some of the results obtained for representations of G a to affine SL 2 -varieties. As in Sect. 6 we identify G a with the unipotent subgroup U ⊂ SL 2 via s → 1 s 0 1 , and G m with the maximal torus T ⊂ SL 2 via t → t 0 0 t −1 . Thus every SL 2 -variety X can be regarded as a G a -variety. These G a -varieties have some very special properties, e.g. the following classical result which was already used in the proof of Theorem 6.1 (see [12,III.3.2]).

Lemma 8.1
Let X be an affine SL 2 -variety and denote by k 2 the standard representation of SL 2 . Then the closed G a -equivariant embedding X → X × k 2 , x → (x, e 1 ), induces an isomorphism X/ /G a ∼ − → (X × k 2 )/ / SL 2 . In particular, the G a -invariants O(X ) G a are finitely generated.

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An immediate consequence is that for every closed embedding X → Y of affine SL 2varieties the induced map X/ /G a → Y / /G a is also a closed embedding.

Proposition 8.2
Let V be a representation of SL 2 and X ⊂ V a closed SL 2 -stable subset.
Proof Since X = V ∩ X and V = V ∪ C σ (Lemma 7.6), we get Therefore, X = S X if and only if X ⊇ V ∩ (X × X ) and (C \C σ ) ∩ (X × X ) ⊂ X . But the latter condition is clearly equivalent to (x 0 + V + ) ∩ X = G a x 0 for all x 0 ∈ X 0 \(X 0 ) σ . ⊓ Example 8.4 Let X := SL 2 /T where T is acting by right multiplication on SL 2 . This variety is the smooth 2-dimensional affine quadric X = V(x z − y 2 + y) ⊂ k 3 , and the quotient map is given by Clearly, X is an SL 2 -variety where the action is induced by left multiplication on SL 2 , and thus a G a -variety. The quotient by G a is A 1 , and the quotient map is given by The plinth ideal is generated by z and is reduced. The plinth variety P X consists of the two

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Moreover, the induced morphisms X\O i → A 1 are both trivial G a -bundles, and so X\O i ≃ A 2 for i = 1, 2. Thus X bd = X , but π : It follows that Since S X ⊂ X × X is the hypersurface defined by f := π • pr 1 −π • pr 2 , the irreducible components of S X have codimension 1 in X × X , hence S X = X .
Example 8.5 Now let us look at Y := SL 2 /N , where N = T ∪ σ T is the normalizer of T . Then σ induces an automorphism of order 2 on X = SL 2 /T commuting with the G aaction, and the automorphism − id on the quotient X/ /G a = A 1 . Thus Y = X/⟨σ ⟩ and Y / /G a = A 1 /{± id} ≃ A 1 . Since σ (O 1 ) = O 2 in the notation of Example 8.4 we see that the plinth variety P Y = π −1 (0) is a single orbit, but the plinth ideal p Y is not prime. Therefore, π : Y → A 1 is a geometric quotient, but not a principal G a -bundle. In this case, Y bd = Y alg = X \P Y , and S Y = Y .

ROBERTS' example
In this section we discuss Roberts' counterexample to Hilbert's fourteenth problem [22]. We assume that char k = 0 and define an action of the additive group G a on A 7 as follows: It corresponds to the locally nilpotent vector field where we use the coordinates O( G a and let π : A 7 → A 7 / /G a := Spec(R) denote the quotient morphism. The x i are invariants, and D(y i ) = x 3 i , hence x 3 i ∈ p A 7 / /G a , and so . This allows to find the following additional invariants: Define the following subalgebras of the ring of invariants R: We then have G a Using a symbolic computation software like Singular [2], it is easy to see that Y 0 := Spec R 0 ⊂ A 6 is the normal hypersurface defined by the equation x 3 1 u 12 +x 3 2 u 13 +x 3 3 u 23 = 0, 123 and that Y := Spec R 1 ⊂ A 9 has dimension 6 and its ideal I (Y ) is generated by the following 5 functions: Remark 9.1 The given relations between the generators of R 0 and R 1 imply that the ideals and 13 , u 23 , β 11 , β 12 , β 13 ]/(u 12 β 3 11 + u 13 β 3 21 + u 23 β 3 31 ).
In particular, is a normal hypersurface of dimension 5.

Lemma 9.2 The variety Y is normal.
Proof Again, using for example Singular [2], one verifies that the ideal x 1 R 1 is radical. Let f ∈ Q(R 1 ) be integral over R 1 , that is, suppose f satisfies an equation where a i ∈ R 1 . Since (R 1 ) x 1 is normal, we have x m 1 f ∈ R 1 for some m ≥ 0. We choose a minimal m with this property. It follows from the equation above that (x m and so (G m ) 3 also acts on A 7 / /G a . As the invariants u i j , β i j are multihomogeneous, (G m ) 3 also acts on Y 0 and Y . The following two propositions collect the main properties of π : A 7 → A 7 / /G a . Most statements follow from what we have done so far. The difficult part is the description of the finite generation ideal f X/ /G a .
Recall that P A 7 ⊂ A 7 denotes the plinth variety, P A 7 / /G a ⊂ A 7 / /G a the plinth scheme (Definition 5.1), and S A 7 ⊂ A 7 × A 7 the separating variety (Sect. 7).
(c) The separating variety S A 7 has two irreducible components:

and so
In particular, (A 7 / /G a ) alg is algebraic.
The inclusion R 1 ⊂ R defines an invariant morphism ϕ : A 7 → Y which factors through the quotient π: (d)φ induces a closed immersion P A 7 / /G a → Y with image in C. In particular, A proof that ϕ is a separating morphism and that (c) holds already appeared in [5,Example 4.2].
Proof Statement (a) holds since (A 7 / /G a ) x i ≃ Y x i by ( * ) above. We have seen in Lemma 9.2 that Y is normal, and the morphismφ is injective on π(A 7 ) sinceφ(π( (c) follows from Theorem 4.3(e), because Y \ϕ( For (d) we have to show that R 1φ → R R/f A 7 / /G a = O(P A 7 / /G a ) is surjective and contains x 1 , x 2 , x 3 in its kernel. For this we use two results which will be proved below. By Proposition 9.3(d) we have f A 7 / /G = √ (x 1 , x 2 , x 3 ). Hence, by Lemma 9.8, we get R/f A 7 / /G a = k[ū 12 ,ū 13 ,ū 23 ], and so R 1 → R/f A 7 is surjective and contains x 1 , x 2 , x 3 in the kernel. ⊓
To prove that O(A 7 ) G a is not finitely generated, Roberts showed in [22,Lemma3] that there exist invariants of the form x i z n + terms of lower z-degree for i = 1, 2, 3 and n ≥ 0. Later, Kuroda proved (see [14,Theorem3.3]) that any set S of such invariants, together with u 12 , u 13 , u 23 , forms a SAGBI-basis for the lexicographic monomial ordering with x 1 ≺ x 2 ≺ x 3 ≺ y 1 ≺ y 2 ≺ y 3 ≺ z. We will improve this statement in Lemma 9.6 below.
Recall that if R is a subalgebra of a polynomial ring, then for a given monomial ordering, a SAGBI-basis is a subset S ⊂ R such that k[LT(S)] = k[LT(R)] where LT(S) denotes the 123 set of leading terms of the polynomials in S (see [21]). Such a basis always generates R.
Proof By induction, it suffices to show that LT( f 2 ) = LT(h) where h ∈ (x 1 , x 2 , x 3 ). But LT( f ), as a monomial in LT(S), must contain a factor of the form x i or x i z since otherwise the multi-degree is congruent to (k, k, k) modulo 3. Hence, LT( f 2 ) contains a factor x i , and so LT( f 2 ) = LT(x i p) for some p ∈ R.
Next we remark that u i j / ∈ p, for all i, j. In fact, if u k i j ∈ (x 1 , x 2 , x 3 )R, then LT(u k i j ) = LT(u i j ) k is a monomial in LT(S 0 ) containing a factor x j which is impossible. Since β in ∈ p it follows that R/p is generated by the (nonzero) images of u 12 , u 13 , u 23 which are algebraically independent, because their multi-degrees are linearly independent. Thus, R/p is a polynomial ring in 3 variables, and p is generated by {β in }. ⊓ Proof of Proposition 9.3(d)-(e) For (d) we already know that x 1 , x 2 , x 3 ∈ f X/ /G a , hence √ (x 1 , x 2 , x 3 ) ⊆ f X/ /G a , and by Lemma 9.8 we have β in ∈ f X/ /G a for all i, n. Now let f ∈ f X/ /G a . Since R = R 0 + (β in )R we can assume that f ∈ R 0 . Since R f is finitely generated there is an N > 0 such that R f = (R N ) f , and so f k β i N+1 ∈ R N for some k > 0 and all i. Hence f k ∈ (x 1 , x 2 , x 3 )R by Lemma 9.7(c), and (d) follows.
Finally, (e) follows from the above and Lemma 9.8. ⊓