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Conjugation on varieties of nilpotent matrices

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of 2-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. Concerning the action on the nilpotent cone, we obtain a generic normal form of the orbits which yields a U-normal form as well, here U is the standard unipotent subgroup. We describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the U-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related.


Introduction
The study of algebraic group actions on affine varieties, especially the "vertical" study of orbits and their closures, and the "horizontal" study of parametric families of orbits and quotients, are a common topic in algebraic Lie theory.
A well-known example is the study of the adjoint action of a reductive algebraic group on its Lie algebra and numerous variants thereof, in particular the conjugacy classes of complex (nilpotent) square matrices.
Algebraic group actions of reductive groups have particularly been discussed elaborately in connection with orbit spaces and more generally algebraic quotients, even though their application to concrete examples is far from being trivial. In case of a non-reductive group, even most of these results fail to hold true immediately.
For example, Hilbert's theorem [11] yields that for reductive groups, the invariant ring is finitely generated; and a criterion for algebraic quotients is valid [13]. In 1958, though, M. Nagata [17] constructed a counterexample of a not finitely generated invariant ring corresponding to a non-reductive algebraic group action, which answered Hilbert's fourteenth problem in the negative.
One exception are algebraic actions of unipotent subgroups that are induced by reductive groups, since the corresponding invariant ring is always finitely generated [13].
We turn our main attention towards algebraic non-reductive group actions that are induced by the conjugation action of the general linear group GL n over C. For example, the standard parabolic subgroups P (and, therefore, the standard Borel subgroup B) and the unipotent subgroup U of GL n are not reductive. It suggests itself to consider their action on the variety N (x) n of x-nilpotent matrices of square size n via conjugation which we discuss in this work.
A recent development in this field is A. Melnikov's study of the B-action on the variety of upper-triangular 2-nilpotent matrices via conjugation [14,15] motivated by Springer Theory. The detailed description of the orbits and their closures is given in terms of socalled link patterns; these are combinatorial objects visualizing the set of involutions in the symmetric group S n . In [5], M. Reineke and the author generalize these results to the Borel-orbits of all 2-nilpotent matrices and describe the minimal, disjoint degenerations corresponding to their orbit closure relations. Furthermore, L. Fresse describes singularities in the upper-triangular orbit closures by translating the group action to a certain group action on Springer fibres (see [7]).
Another recent outcome is L. Hille's and G. Röhrle's study of the action of P on its unipotent radical P u , and on the corresponding Lie algebra p u (see [12]). They obtain a criterion which varifies that the number of orbits is finite if and only if the nilpotency class of P u is less or equal than 4. This result is generalized to all classical groups G.
Given a semi-simple Lie algebra g and its Lie group G, D. Panyushev considers the adjoint action in [18] and shows that, given a nilpotent element e ∈ g\{0}, the orbit G.e is spherical if and only if (ad e ) 4 = 0. The notion of sphericity translates to G.e admitting only a finite number of Borel-orbits, due to M. Brion [6].
In this work, we make use of a translation of the classification problem of the P-orbits in N (x) n to the description of certain isomorphism classes of representations of a finitedimensional algebra in Section 3. By making use of this translation, we describe the P-orbits in N (2) n as well as their closures in detail. Furthermore, we specify all minimal degenerations in Section 4. This particular action admits only a finite number of orbits and, by considering P-actions on N (x) n , we find a criterion as to whether the action admits a finite number of orbits in Section 5.
By considering the nilpotent cone N, we generalize the generic B-normal form given in [10,5] to arbitrary upper-block parabolic subgroups in Section 6. We describe B-semiinvariants that generate the ring of all B-semi-invariants and -as a direct consequence -find U-invariants that generate the U-invariant ring in Section 7. The latter will be made use of to discuss the U-invariant ring in more detail in Section 8 by proving a quotient criterion and discussing a toric variety closely related to the algebraic quotient of N by U.

Theoretical background
We denote by K ≔ C the field of complex numbers and by GL n ≔ GL n (K) the general linear group for a fixed integer n ∈ N regarded as an affine variety. In order to fall back on certain results later on in the article, we include basic knowledge about (semi-) invariants and quotients [13,16]; and about the representation theory of finitedimensional algebras [1].
3. on a non-empty open subset Y 0 ⊆ Y the fibre π −1 (y) contains exactly one closed orbit for each y ∈ Y 0 , then π is an algebraic G-quotient of X.
In case G is not reductive, there are counterexamples of only infinitely generated invariant rings (see [17]

Toric varieties
Since our considerations will involve the notion of a toric variety, we discuss it briefly. For more information on the subject, the reader is referred to [8].
A toric variety is an irreducible variety X which containes (K * ) n as an open subset, such that the action of (K * ) n on itself extends to an action of (K * ) n on X.
Let N be a lattice, that is, a free abelian group N of finite rank. By M := Hom Z (N, Z) we denote the dual lattice, together with the induced dual pairing _, _ . Consider the vector space and if there is a finite set S ⊆ N that generates σ, that is, Given a strongly convex rational polyhedral cone σ, we define its dual by and its corresponding additive semigroup by S σ := σ ∨ ∩ M, which is finitely generated due to Gordon's lemma (see [8]). Note that if σ is a maximal dimensional strongly convex rational polyhedral cone, then σ ∨ is one as well. We associate to it the semigroup algebra KS σ and obtain an affine toric variety Spec KS σ .
3. An affine toric variety X is isomorphic to Spec KS σ for some strongly convex rational polyhedral cone σ if and only if X is normal.

Representation theory of finite-dimensional algebras
A finite quiver Q is a directed graph Q = (Q 0 , Q 1 , s, t), such that Q 0 is a finite set of vertices and Q 1 is a finite set of arrows, whose elements are written as α : s(α) → t(α).
The path algebra KQ is defined as the K-vector space with a basis consisting of all paths in Q, that is, sequences of arrows ω = α s . . . α 1 , such that t(α k ) = s(α k+1 ) for all k ∈ {1, . . . , s − 1}; formally included is a path ε i of length zero for each i ∈ Q 0 starting and ending in i. The multiplication is defined by where ωω ′ is the concatenation of paths ω and ω ′ .
We define the radical rad(KQ) of KQ to be the (two-sided) ideal generated by all paths of positive length; then an arbitrary ideal I of KQ is called admissible if there exists an integer s with rad(KQ) s ⊂ I ⊂ rad(KQ) 2 .
where the M i are K-vector spaces, and the M α are K-linear maps.
For a representation M and a path ω in Q as above, we denote These definitions yield certain categories as follows: We denote by rep K (Q) the abelian K-linear category of all representations of Q and by rep K (Q, I) the category of representations of Q bound by I; the latter is equivalent to the category of finite-dimensional KQ/I-representations.
Given a representation M of Q, its dimension vector dimM ∈ NQ 0 is defined by (dimM) i = dim k M i for i ∈ Q 0 . Let us fix a dimension vector d ∈ NQ 0 , then we denote by rep K (Q, I)(d) the full subcategory of rep K (Q, I) which consists of representations of dimension vector d. One standard technique to calculate the Auslander-Reiten quiver is the knitting process (see, for example, [1,IV.4]). In some cases, the Auslander-Reiten quiver Γ(Q, I) can be calculated by using covering techniques (see [9] or [3]).
By defining the affine space R d (Q) := α: i→ j Hom K (K d i , K d j ), one realizes that its points m naturally correspond to representations M ∈ rep K (Q)(d) with M i = K d i for i ∈ Q 0 . Via this correspondence, the set of such representations bound by I corresponds to a closed subvariety R d (Q, I) ⊂ R d (Q). Let us denote by add Q the additive category of Q with objects O(i) corresponding to the vertices i ∈ Q 0 and morphisms induced by the paths in Q. Since every representation M ∈ rep K (Q) can naturally be seen as a functor from add Q to Mod K, we denote this functor by M as well. Let φ : O(i) y i be an arbitrary morphism in add Q and consider d ∈ NQ 0 , such that i∈Q 0 x i · d i = i∈Q 0 y i · d i . An induced so-called determinantal semi-invariant is given by where m ∈ R d (Q) and M ∈ rep K (Q)(d) are related via the above mentioned correspondence. The following theorem (see [19]) is due to A. Schofield

Translation to a representation-theoretic setup
We fix a parabolic subgroup P of GL n of block sizes (b 1 , . . . , b p ).
We define Q p to be the quiver and A(p, x) ≔ KQ p /I x to be the finite-dimensional algebra, where I x ≔ (α x ) is an admissible ideal. We fix the dimension vector and formally set b 0 = 0. As explained in Section 2.3, the algebraic group GL d P acts on R d P (Q p , I x ); the orbits of this action are in bijection with the isomorphism classes of representations in rep K (Q p , I x )(d P ).
Let us define rep inj K (Q p , I x )(d P ) to be the full subcategory of rep K (Q p , as in Section 2.3. In order to describe the orbit closure The following lemma is a slightly generalized version of [5,Lemma 3.2]. The proof is similar, though. n . Thus, there exists a bijection Φ between the set of P-orbits in N (x) n and the set of which sends an orbit P.N ⊆ N (x) to the isomorphism class of the representation This bijection preserves orbit closure relations, dimensions of stabilizers (of single points) and codimensions.
Due to considerations of different parabolic subgroups and nilpotency degrees, the classification of the corresponding isomorphism classes of representations differs wildly.

P-conjugation on N (2)
n Let us consider the action of P on the variety N (2) n of 2-nilpotent n × n-matrices. As the theorem of W. Krull, R. Remak and O. Schmidt states, every representation in rep K (Q p , I 2 ) can be decomposed into a direct sum of indecomposables, which is unique up to permutations and isomorphisms. Following [5,Theorem 3.3], the following lemma classifies the indecomposables in rep K (Q p , I 2 ).

Lemma 4.1. Up to isomorphisms, the indecomposable representations in
are the following: Here, e 1 and e 2 are the standard coordinate vectors of K 2 and α · e 1 = e 2 , α · e 2 = 0.
An enhanced oriented link pattern of type (b 1 , . . . , b p ) is an oriented graph on the vertices {1, . . . , p} together with a (possibly empty) set of of dots at each vertex, such that the sum of the numbers of sources, targets and dots at every vertex i equals b i . Clearly, an enhanced oriented link pattern of a fixed type is far from being unique.
For example, an enhanced oriented link pattern of type (3, 2, 6, 2, 5) is given by Theorem 4.2. There are natural bijections between

Moreover, if the isomorphism class of M corresponds to a matrix N under this bijection, the orbit
The proof is similar to the proof of [5,Theorem 3.4]. Note that the multiplicity of the indecomposable V i is obtained as the number of dots at the vertex i which we call "fixed vertices". The multiplicity of the indecomposable U i, j is given as the number of arrows j → i. We define eolp(X) to be the enhanced oriented link pattern corresponding to both the isomorphism class of X ∈ rep inj K (Q p , I 2 )(d P ) and the P-orbit of X ∈ N (2) n . An oriented link pattern of size n is an enhanced oriented link pattern of type (1, . . . , 1). Thus, every vertex is incident with at most one arrow. The concrete classification of the Borel-orbits is then given by the oriented link patterns of size n and is easily obtained from Theorem 4.2 (see, for the detailed proof, [5, Theorem 3.4]). As before, we define olp(X) to be the oriented link pattern corresponding to both the isomorphism class of These dimensions are calculated for indecomposable representations in rep K (Q p , I 2 ) in [5,Lemma 4.2].
In order to prove an easy description of the parabolic orbit closures in N (2) n in terms of (enhanced) oriented link patterns, we discuss how the dimensions of Proposition 4.3 are linked with these.
Let N ∈ N (2) n be a 2-nilpotent matrix that corresponds to the representation M via the bijection of Lemma 3.1.

Borel-orbit closures
Let M and M ′ be two representations in rep K (Q n , I 2 ) of the same dimension vector d.
Since the correspondence of Lemma 3.1 preserves orbit closure relations, we know that M ≤ deg M ′ if and only if the corresponding 2-nilpotent matrices, denoted by N = The following theorem can be found in [5,Theorem 4.3].
The key to calculating all minimal degenerations is obtained by the following proposition (see [5,Corollary 4.5]).

where U and V are indecomposables and there exists an exact sequence
A method to construct all orbits contained in a given orbit closure is described in [5,Theorem 4.6], since Proposition 4.7 "localizes" the problem to sequences of changes at at most four vertices of the corresponding oriented link pattern. All these minimal, disjoint degenerations are explicitly listed (in terms of oriented link patterns as well) in [5,Theorem 4.6].
Consider an arbitrary minimal, disjoint degeneration D < mdeg D ′ in rep inj K (Q n , I 2 ). To classify the minimal degenerations in rep inj We give an explicit criterion as to whether this degeneration is minimal.
Proof. If the degeneration is obtained from an extension as in Proposition 4.7, the claim follows from [2, Theorem 4] (see the exact argumentation in [4,Theorem 3.3.11]).
Assume that the minimal, disjoint degeneration D < mdeg D ′ is given by U s,t < mdeg U t,s , that is, by the only minimal, disjoint degeneration not obtained from an extension.
The theorem then reads as follows: If s k < t and s < l < t (or s < k < t and l > t, respectively), then the degeneration U t,s ⊕ U k,l < deg U s,t ⊕ U k,l is not minimal, since  Proof of Claim 2. Let t < k and s < l < t for two integers k and l.
First, we assume that U k,l is a direct summand of M, but not a direct summand of L. Since M < mdeg L, the indecomposable U k,l must be changed by some minimal, disjoint part of the degeneration. The only possibilities for a change like that are the following: As has been shown in claim 1, every indecomposable U i, j with j < s, or with j > t and i < s is either a direct summand of M, L and M ′ or a direct summand of none of them. Thus, k ′ > t and if k < k ′ and l ′ < l, we obtain The indecomposable U l ′ ,k ′ can only be a direct summand of M if l ′ > t or k ′ < s, or if l ′ < s and k ′ > t. As has been shown in claim 1, every indecomposable U i, j with j < s, or with j > t and i < s is either a direct summand of M, L and M ′ or a direct summand of none of them.
Thus, l ′ > t and the only cases possible are l < l ′ < k ′ < k and l < k ′ < k < l ′ . We immediately obtain [U k ′ ,l , M] < [U k ′ ,l , L], a contradiction. 2nd case: The indecomposable U k,l ′ is a direct summand of L, such that l l ′ . 2.1. The minimal, disjoint part is U k,l ⊕ V l ′ < mdeg U k,l ′ ⊕ V l , such that l < l ′ : The indecomposable V l ′ can only be a direct summand of M if l ′ < s or l ′ > t. Thus, l ′ > t and we obtain [U t,l , M] < [U t,l , L], a contradiction.
As has been shown in claim 1, every indecomposable U i, j with j < s, or with j > t and i < s is either a direct summand of M, L and M ′ or a direct summand of none of them, thus, l ′ > t. But then we obtain [U t,l , M] < [U t,l , L] , a contradiction. 3rd case: The indecomposable U l,k is a direct summand of L.
Of course, if s = 1 and t = n > 2, no representation W as given in the assumption can exist at all, a contradiction.
The assumption that U k,l is a direct summand of L, but not a direct summand of M can be contradicted by a similar argumentation.
Claim 1 and claim 2 show that all arrows l → k with b k,l (M) = b k,l (M ′ ) and k, l {s, t} coincide in olp(M), olp(L) and olp(M ′ ). The minimal, disjoint piece of the degeneration D ⊕ W < mdeg L, therefore, has to be one of the following three.
• U t,s < mdeg U s,t : Then L M ′ , a contradiction to the assumption L < deg M ′ .
and therefore L ≮ deg M ′ , a contradiction.
Since we obtain a contradiction in each case, the degeneration M < deg M ′ is minimal.
Note that in the setup of The concrete minimal degenerations are obtained easily from Proposition 4.3. Furthermore, each minimal degeneration is of codimension 1 (which is, as well, clear from the theory of spherical varieties, see [6]).

Parabolic orbit closures
In case of the action of P, we describe all minimal, disjoint degenerations analogously to [5,Theorem 4.6]. Then it either appears in [5,Theorem 4.6] or in one of the following chains.
These minimal, disjoint degenerations yields concrete descriptions of the orbit closures in terms of enhanced oriented link patterns right away.

A finiteness criterion
We consider the P-action on N (x) n and prove a criterion as to whether the action admits finitely many or infinitely many orbits. We call the parabolic subgroup P maximal, if it is given by 2 blocks (b 1 , b 2 ).

Theorem 5.1. There are only finitely many P-orbits in N (x)
n if and only if x ≤ 2, or P is maximal and x = 3.
Proof. If x = 2, our considerations in Section 4 yield finiteness for every parabolic subgroup P.
If P is maximal of block sizes (b 1 , b 2 ) and x = 3, then in order to calculate a system of representatives of the indecomposable representations, it suffices to calculate those of the quiver with the relation α 3 α 2 α 1 = 0 which follows from the before mentioned covering theory of quivers. These can be calculated by using the knitting process and the quiver in Figure 1 is obtained. Note that we directly delete zero rows in the dimension vectors.
There are only finitely many isomorphism classes of indecomposable representations, thus the translation to the P-orbits in N If P is a maximal parabolic subgroup of block sizes (x, y), then the action of P on N (4) n admits infinitely many orbits: 1. If x = s + 2 ≥ 2 and y = t + 2 ≥ 2 for s, t ≤ 0, then the matrices where for λ ∈ K * , induce a 1-parameter family of pairwise non-P-conjugate matrices.
2. If (without loss of generality) x = 1 and y = n − 1, then for λ ∈ K * , the matrices induce a 1-parameter family of pairwise non-P-conjugate matrices.
Note that the algebra A(p, x) is either of finite or of wild representation type, but never of infinite tame representation type.

Generic normal forms in the nilpotent cone
We discuss the P-action on the nilpotent cone N := N (n) n now and introduce a generic normal form. We, thereby, generalize a generic normal form for the orbits of the Borelaction which is introduced in [5,10].
Let V be an n-dimensional K-vector space and denote the space of partial p-step flags of dimensions d by F d P (V), that is, F d P (V) contains flags Proof. By [5, Theorem 5.1], we find a basis u 1 , . . . , u n of V that is adapted to F * and such that ϕ (u x ) = u x+1 mod u x+2 , . . . , u n .
It is clear by the theorem of the Jordan normal form that we can modify this basis, such that One can now verify the existence of the sought basis by adapting the given basis accordingly.
The normal form is sketched in Figure 2 where the block sizes are those of the parabolic subgroup P. As a direct consequence, the affine space

Generation of (semi-) invariant rings
From now on, we consider the action of the Borel subgroup B and the unipotent subgroup U on the nilpotent cone N. We define (semi-) invariants which generate the corresponding ring of (semi-) invariants (as we will see in Theorem 7.2). Let us start by defining those Borel-semi-invariants introduced in [5, Proposition 5.3].
Given i ∈ {1, . . . , n}, we denote by ω i : B → G m the character which is defined by ω i (g) = g i,i ; the ω i form a basis for the group of characters of B.
Let us fix integers s, t ∈ N. For i ∈ {1, . . . , s} and j ∈ {1, . . . , t}, we fix integers a i , a ′ j ∈ {1, . . . , n} with a 1 + . . . + a s = a ′ 1 + . . . + a ′ t =: r and polynomials P i, j (x) ∈ K[x]. Let N ∈ N, then for all such i and j we consider the submatrices P i, j (N) (a i ,a ′ j ) ∈ K a i ×a ′ j and form the r × r-block matrix Note that the function f P is also a U-invariant regular function on N.

Theorem 7.2. The semi-invariant ring K[N] B
* is generated by the semi-invariants of Proposition 7.1.
] induces a surjection on the corresponding semi-invariant rings, since GL d B is reductive. Furthermore, the codimension of is greater or equal than 2, which yields the claim. (Q n , I x ), corresponds to one of the B-semi-invariants of Proposition 7.1.
Let us fix an arbitrary morphism in add Q, say φ : Then, by Section 2.3, we obtain a determinantal semi-invariant f φ .
The homomorphism spaces P( j, i) between two objects O( j) and O(i) in add Q are generated as K-vector spaces by The morphism φ is given by a n i=1 y i × n j=1 x j -matrix H with entries being morphisms between objects in add Q. We can view the matrix H as an n × n block matrix H = (H i, j ) 1≤i, j≤n with H i, j ∈ K y i ×x j for i, j ∈ {1, . . . , n}. Then Given an arbitrary matrix N ∈ N, we reconsider the representation M N defined in Theorem 3.1. Since GL d B acts transitively on R inj d B (Q ′ ) with Q ′ being the linearly oriented quiver of Dynkin type A n , we can examine the restricted semi-invariant on these representations M N .
The B-semi-invariant of N associated to f φ via the translation of Lemma 3.1 is given by is given as a block matrix where each block is again a block matrix. The blocks of M N i, j are given by , if j ≤ i = n; such that E (i) ∈ K i×i is the identity matrix. Note that if i, j ∈ {1, . . . , n} and i < n, then M N i, j = M N ′ i, j =: M i, j for every pair of matrices N, N ′ ∈ N. We can without loss of generality assume y 1 = . . . = y n−1 = 0 which can, for example, be seen by induction on the index i of y i . This assumption is not necessary for the proof, but will shorten the remaining argumentation. Let us define Let us denote P ≔ a, a ′ , P (k,l) j j,k,l and let N ∈ N; it suffices to show f P (N) = f φ (N):

About the algebraic U-quotient of the nilpotent cone
We have seen that the U-invariant ring K[N] U is spanned by the functions defined in Proposition 7.1. At least for the cases n = 2, 3 the quotient criterion which we prove (in a more general setup) in the next subsection helps to provide the explicit structure of these rings.

A quotient criterion
Let G be a reductive algebraic group and U be a unipotent subgroup. Then U acts on G by right multiplication and Lemma 2.2 states that the U-invariant ring K[G] U is finitely generated as a K-algebra. Thus, an algebraic U-quotient of G, namely G/ /U ≔ Spec K[G] U , exists together with a dominant morphism π G/ /U : G → G/ /U which is in general not surjective. Note that there is an element e ∈ G/ /U, such that π G/ /U (g) = ge for all g ∈ G.
The group G acts on G/ /U by left multiplication. Let X be an affine G-variety and consider the diagonal operation of G on the affine variety G/ /U × X; we consider the natural G-equivariant morphism ι : G be the associated algebraic G-quotient, then we obtain a morphism The morphism ρ induces an isomorphism ρ * : Thus, X/ /U (G/ /U × X)/ /G and Let Y be an affine G-variety and let µ ′ : G/ /U × X → Y be a G-invariant morphism, together with a dominant U-invariant morphism of affine varieties In this setting, we obtain the following criterion for µ to be an algebraic U-quotient.
Then µ is an algebraic U-quotient of X, that is, Y X/ /U.
The morphism µ ′ separates the G-orbits in G/ /U × X generically (that is, in Y U ): The morphism µ ′ restricts to a surjection G.({e} × X U ) → Y U , furthermore, the algebraic quotient π ′ is surjective and there exists a morphismμ ′ : (G/ /U ×X)/ /G → Y, such thatμ ′ •π ′ = µ ′ . Then Y U ⊆ im(μ ′ ) and, since each fibre of π ′ contains exactly one closed G-orbit, we have shown that generically each fibre of µ ′ contains a unique closed orbit.
Thus, Theorem 2.1 yields that π : G/ /U × X → Y is an algebraic G-quotient. Since f i and g i correspond to each other via the isomorphism ρ * : We are now able to give explicit descriptions of algebraic U-quotients of the nilpotent cone in case n equals 2 or 3.
The affine variety Y is normal as the product of A 1 and a normal toric affine variety induced by the strongly convex rational polyhedral cone The morphism µ separates the U-orbits in the open subset N U ⊆ N as can be proved by a direct calculation.

Toric invariants
As the case n = 3 suggests, there is a toric variety closely related to N/ /U.
By considering a special type of U-invariants, so-called toric invariants, we define a toric variety X together with a dominant morphism N/ /U → X, such that the generic fibres are affine spaces of the same dimension.
Given a matrix H = (x i, j ) i, j ∈ H U , we denote x i ≔ x i+1,i and define its toric part H tor ∈ K n×n by (H tor ) i, j ≔ x i , if i = j + 1; 0, otherwise.
Let f 0 be a invariant, given by the data .
We call f toric if f (H) = f (H tor ) for every matrix H ∈ H U and sum-free if its block sizes a 1 , . . . , a s and a ′ 1 , . . . , a ′ t do not share any partial sums, that is, i∈I a i i ′ ∈I ′ a ′ Denote by S the set of tuples (h 1 , . . . , h n−1 ) ∈ N n−1 that arise in this way from a minimal set of generating toric invariants and denote σ ≔ Cone(S ).
Let N be the lattice Z n−1 , then σ is generated by the finite set S ⊂ Z n−1 and fulfills σ ∩ (−σ) = {0}, therefore, σ as well as σ ∨ are strongly convex rational polyhedral cones of maximal dimension. The variety X = Spec K[N] U tor Spec K[S σ ∨ ], thus, is a normal toric variety by Lemma 2.3.
Let T ⊂ GL n be the torus of diagonal matrices. There is a natural action τ of T on the U-invariant ring of N as follows: Proof. The morphism p is clearly dominant and T -equivariant due to our considerations above.
Let x ′ ∈ X ′ , then p −1 (x) ⊆ i(H U ), since every determinant det i for i ∈ {1, . . . , n − 1} is a toric invariant. If x ′ ∈ X ′ , none of these determinants vanishes on x ′ and Section 6, therefore, yields p −1 (x ′ ) ⊆ i(H U ). Since the orbits in N U are separated by certain U-invariants and since H U A D × X ′ , the claim p −1 (x) A D follows.
There is a morphism q : N/ /U → A D as well, such that the composition Proof. The morphism (p, q) is dominant, since A D × X ′ ⊆ im(p, q) ⊆ A D × X.
The morphism (p, q) is birational, since (p, q) is dominant and generically injective: the fibre (p, q) −1 (y) contains exactly one element for every y ∈ A D × X ′ , since the U-orbits can be separated in A × X ′ . More straight forward, (p, q) restricts to an isomorphism i(H U ) A D × X ′ .