A resolution of singularities for the orbit spaces G n,2 / T n
aa r X i v : . [ m a t h . A T ] S e p A resolution of singularitiesfor the orbit spaces G n, /T n Victor M. Buchstaber and Svjetlana Terzi´cSeptember 4, 2020
Abstract
The problem of the description of the orbit space X n = G n, /T n for thestandard action of the torus T n on a complex Grassmann manifold G n, iswidely known and it appears in diversity of mathematical questions. A point x ∈ X n is said to be a critical point if the stabilizer of its corresponding orbitis nontrivial. In this paper, the notion of singular points of X n is introducedwhich opened the new approach to this problem. It is showed that for n > the set of critical points Crit X n belongs to our set of singular points Sing X n ,while the case n = 4 is somewhat special for which Sing X ⊂ Crit X , butthere are critical points which are not singular.The central result of this paper is the construction of the smooth man-ifold U n with corners, dim U n = dim X n and an explicit description ofthe projection p n : U n → X n which in the defined sense resolve all sin-gular points of the space X n . Thus, we obtain the description of the orbitspace G n, /T n combinatorial structure. Moreover, the T n -action on G n, is a seminal example of complexity ( n − - action. Our results demon-strate the method for general description of orbit spaces for torus actions ofpositive complexity. MSC 2000: 57S25, 57N65, 53D20, 14M25, 52B11, 14B05; keywords: Grassmann manifold,torus action, chamber decomposition, orbit space, universal space of parameters Introduction
The studying of the orbit space X n = G n, /T n of a complex Grassmann man-ifold G n, of the two-dimensional complex planes in C n by the standard actionof the compact torus T n is of wide mathematical interest for a while from manyaspects: of algebraic topology, algebraic geometry, theory of group actions, ma-troid theory, combinatorics. This interest was stimulated by the works Gel’fand-Serganova [11], Goresky-MacPherson [12], Gel’fand-MacPherson [10] in whichthe ( C ∗ ) n -action on general Grassmannian G n,k , < k < n − being an exten-sion of T n -action, was studied. The results of these papers suggest that the case k = 2 needs to be studied taking into account the specialty of the spaces G n, .A little later Kapranov [13] related T n -action on G n, with the notion of Chowquotient from algebraic geometry for which he showed to be isomorphic to theGrotendick-Knudsen moduli space M ,n of stable n - pointed rational curves ofgenus zero. These works attracted a lot attention which resulted in series of paperson related subjects, [16], [17], [22] etc.The complex projective space C P n − = G n, with the standard action of thealgebraic torus ( C ∗ ) n and the induced action of the compact torus T n ⊂ ( C ∗ ) n is the key object in toric geometry and toric topology respectively, see [5]. It iswell known that the orbit space G n, /T n can be identified with the simplex ∆ n − which is the standard example of a smooth manifold with corners. Our studyingof an orbit space X n = G n, /T n with its canonical moment map G n, → X n → ∆ n, , where ∆ n, is a hypersimplex has been motivated by the natural problem ofextension of the methods of toric topology to the case of torus actions of positivecomplexity. More recently in [6] and [7] the orbit spaces X = G , /T and X = G , /T are explicitly described, while an extensive general study of T n -action on G n,k for k ≥ is done in [7], [8] in the context of the theory of (2 n, k ) -manifolds.It is proved in [6] that the space X is homeomorphic to the sphere S . Accordingto [24] the sphere S has a unique smooth structure. Nevertheless, the space X is a topological sphere, that is it is not possible to introduce a smooth structure on X such that the natural projection G , → G , /T is a smooth map. Precisely,it is shown in [6] that S has cone-like singularities at the points with non-trivialstabilizer. In this case the non-trivial stabilizer has the dimension , or and2omplexity of a singularity grows together with the dimension of its stabilizer.The space X is no longer a manifold since it is computed in [7], see also [25] thatthe nontrivial homology groups for X are Z in dimensions and , while it is Z in dimension .A point x ∈ X n is said to be a critical point if its corresponding T n -orbit in G n, has a non-trivial stabilizer. As it is proved in [7], [8] a point x ∈ X n is a criticalpoint if and only if a point from its T n -orbit is a singular point of the smoothstandard moment map µ n, : G n, → ∆ n, in the sense of mathematical analysis,where ∆ n, is considered as a smooth manifold with corners.On the other hand for any < k < n there is the decomposition of G n,k into thestrata, in which a stratum W σ consists of those ( C ∗ ) n - orbits which have the samemoment map image, that is an interior of the polytope P σ ⊂ ∆ n,k . A polytope P σ we call an admissible polytope for W σ . The orbit space F σ = W σ / ( C ∗ ) n is calleda space of parameters of a stratum W σ and it holds W σ /T n ∼ = ◦ P σ × F σ . In par-ticular the main stratum W is a stratum whose admissible polytope is the wholehypersimplex ∆ n,k and its space of parameters we denote by F n,k . The main stra-tum is an open dense set in G n,k which implies that the space ◦ ∆ n,k × F n,k is anopen dense set in the orbit space G n,k /T n . The T n - action on G n,k defines thesemi-upper continuous function from G n,k to the set S ( T n ) of all toral subgroupsin T n , which to any point L ∈ G n,k assigns its stabilizer T L . We proved in [7]that this function is constant T σ on each stratum W σ which implies that the torus T σ = T n /T σ acts freely on W σ . The notions of the strata, corresponding ad-missible polytopes and spaces of parameters have already been present in someforms known in the literature. As we realized these notions are not enough for thedescription of an orbit space G n,k /T n . Therefore, in our papers [7], [8] we intro-duced the new notions of universal space of parameters F n,k and virtual spaces ofparameters ˜ F σ ⊂ F n,k for the strata W σ . The key properties of these notions arethat F n,k is a compactification of the space of parameters F n,k of the main stratumand that F n,k = ∪ σ ˜ F σ . Moreover, there exists the projection p σ : ˜ F σ → F σ forany σ . We explicitly describe F n = F n, in [9] and show that it is a smooth com-pact manifold which can be identified with the Chow quotient G n, //T n definedby Kapranov [13].We define a point x ∈ G n, /T n to be a singular point if for the stratum W σ suchthat x ∈ W σ /T n , the space of parameters F σ is not homeomorphic to the virtualspace of parameters ˜ F σ . It will be shown in the paper that this condition can beformulated in terms of Pl¨ucker coordinates as well.3or n ≥ all critical points of X n belong to the set of singular points, while thecase n = 4 is somewhat special. The main goal of this paper is to construct a smooth manifold with corners U n which functorially resolves the singular points of an orbit space X n being at thesame time compatible with the combinatorial structure of X n described below.Let Y n = X n \ Sing X n and note that Y n is an open, dense set in X n which is amanifold. By the resolution of singularities we mean that U n is a such manifoldfor which there exists a projection p : U n → X n such that for an open, densesubmanifold V n = p − ( Y n ) ⊂ U n the map p : V n → Y n is a diffeomorphism.Note that the sequences of embeddings of the spaces C ⊂ C ⊂ . . . ⊂ C n ⊂ C n +1 ⊂ . . . and the groups T ⊂ T ⊂ . . . ⊂ T n ⊂ T n +1 ⊂ . . . define thesequence of embeddings of the Grassmann manifolds G , ⊂ G , ⊂ . . . G n, ⊂ G n +1 , ⊂ . . . and their orbit spaces X ⊂ X ⊂ X n ⊂ X n +1 ⊂ . . . . We show thatour construction has functorial property meaning that it produces the sequence ofembeddings of the smooth manifolds with corners which accordingly resolve thesingular points of the orbit spaces X n .In the case of the toric manifolds the interiors of admissible polytopes do notintersect. The difficulties in the description of the orbit spaces G n,k /T n is causedby the fact that there are, as a rule, admissible polytopes whose interiors havenon-empty intersectionIn the paper of Goresky-MacPherson [12] it was suggested the decomposition of ∆ n,k into disjoint union of chambers C ω . The chambers are obtained by the inter-sections of the interiors of all admissible polytopes, that is C ω = ∩ σ ∈ ω ◦ P σ , suchthat C ω ∩ ◦ P σ = ∅ if σ ω . They pointed that for any x, y ∈ C ω the preimages or-bit spaces µ − ( x ) /T n and µ − ( y ) /T n are homeomorphic, that is homeomorphicto some compact topological space F ω . Note that F ω has the canonical decom-position as F ω = ∪ σ ∈ ω F σ . The difficulties with the description of the orbit space G n,k /T n - structure is caused by the fact that preimages F ω and F ω are not ingeneral homeomorphic for ω = ω and one needs to make them correspondent.We resolve this problem for the Grassmannians G n, by showing that there exist asmooth manifold F n , that is a universal space of parameters, and the continuousprojection G : U n = ∆ n, × F n → X n , see Theorem 9 and Theorem 10. The pro-4ection G defines on U n = ∆ n, × F n an equivalence relation which we explicitlydescribe in terms of the chamber decomposition of ∆ n, and the correspondingdecompositions of the manifold F n . In the result we obtain that the orbit space X n is a quotient space of a smooth manifold with corners U n = ∆ n, × F n bythis equivalence relation. Our construction is based on two main inputs: first weshow that the decomposition of ∆ n, indicated by Goresky-MacPherson can bedescribed in terms of some special hyperplane arrangement; second we prove thatfor any point x ∈ ∆ n, the union ˜ F x = ∪ ˜ F σ of all virtual spaces of parameters ˜ F σ ,which correspond to the admissible polytopes P σ such that x ∈ ◦ P σ coincide withthe universal space of parameters, that is ˜ F x = F n . In the result we obtain the re-quired projection U n = ∆ n, × F n → X n . The smooth manifold with corners U n is a resolution of singularities of X n which is functorially related to the sequenceof the natural embeddings of these orbit spaces and which is compatible with thestructure of the above chamber decomposition of the hypersimplex ∆ n, .Moreover, it is defined the action of the symmetric group S n on X n by the actionof the normalizer N ( T n ) in U ( n ) on G n, = U ( n ) / ( U (2) × U ( n − . The group S n acts as well on ∆ n, by the permutation of the vertices, this action preservesthe chamber decomposition of ∆ n, and the induced moment map ˆ µ n,, : X n → ∆ n, is S n - equivariant. Using this one can define S n -action on U n and the map G : U n → X n , which resolves the singular points of X n is S n -equivariant.We want to point that our results may have application in the study of positivecomplexity torus actions which has been recently extensively developing, see forexample [1], [3], [14], [26]. In that context the theory of spherical manifoldsincluding theory of homogeneous spaces of compact Lie groups provides numer-ous examples of positive complexity torus actions, [3]. The interesting examplesarise as well in the theory of symplectic manifolds with Hamiltonian torus actions,[15]. In the recent time many non-trivial results about positive complexity torusactions are obtained by the methods of equivariant algebraic topology.In addition, in the focus of recent studies are the description of torus orbit closuresfor different varieties, see [20], [21], [23]. Our geometric analytical description ofthe admissible, that is matroid, polytopes for G n, may lead to the results in thisdirection. 5 T n - equivariant automorphisms of G n,k We first analyse the relations between the T n -equivariant automorphisms of G n,k and the standard moment map µ n,k : G n,k → R n .Let G n,k be a complex Grassmann manifold of k -dimensional complex planesin C n . It is known [4] that the group of holomorphic automorphisms Aut G n,k is isomorphic to the projective group P U ( n ) if n = 2 k , while for G k,k it isisomorphic to Z × P U ( n ) , where n = 2 k . The group Z is defined by the dualityisomorphism. The duality isomorphism for G k,k we obtain from the standarddiffeomorphism c n,k : G n,k → G n,n − k : denote by l : C n → ( C n ) ∗ the canonicalisomorphism, where ( C n ) ∗ is the dual space for C n , let L ∈ G n,k and put L ′ = { λ ∈ ( C ∗ ) n | λ ( v ) = 0 for any v ∈ L } . Then it is defined c n,k ( L ) = l − ( L ′ ) andfor n = 2 k it gives one more non-linear isomorphism for G k,k .Further, the symmetric group S n acts on C n by permuting the coordinates, so any s ∈ S n produces the automorphism s : G n,k → G n,k . It follows that S n is asubgroup of Aut G n,k .Let us consider the canonical action of the compact torus T n on G n,k and let S bethe diagonal circle in T n . Then T n − = T n /S acts effectively on G n,k and it is amaximal torus in P U ( n ) . The normalizer of T n − in P U ( n ) is T n − ⋊ S n . Thecanonical diffeomorphism c n,k : G n,k → G n,n − k is equivariant for the canonicaltorus action as well. We summarize: Lemma 1.
The subgroup of Aut G n,k which contains those elements that commuteswith the canonical T n -action on G n,k is T n − ⋊ S n for n = 2 k and it is Z × ( T n − ⋊ S n ) for n = 2 k . Let W ⊂ G n,k /T n be the set of fixed point for the considered T n -action on G n,k .We find useful to note the following: Proposition 1.
Let H be a subgroup of those holomorphic automorphisms for G n,k for which the set W is invariant. Then H = T n − ⋊ S n for n = 2 k and i H = Z × ( T n − ⋊ S n ) for n = 2 k .Proof. We need to prove that if f ∈ H then f commutes with the canonical T n -action on G n,k . The fixed points for the canonical action of T n on G n,k are k -dimensional coordinate subspaces in C n . Let v be a coordinate vector. There are (cid:0) n − k − (cid:1) coordinates k -subspaces which contains v . Let L be a such subspace and6 ∈ P U ( n ) which leaves W invariant. For the linear isomorphism ˆ f representing f it holds that ˆ f ( L ) is a k -dimensional coordinate subspace defined by z i = . . . = z i k = 0 for some ≤ i < . . . < i k ≤ n , which implies that the coordinates of ˆ f ( v ) indexed by i , . . . , i k are equal to zero. Since there are at least n − different k -dimensional coordinate subspaces which contains v it follows that ˆ f ( v ) hasto be, up to constant, a coordinate vector. Therefore the isomorphism ˆ f , up toconstants, permutes the coordinate vectors. It implies that ˆ f commutes with T n -action on C n , so f commutes with the induced T n -action on G n,k . When k = 2 n ,the nontrivial map f ∈ Z any k -dimensional coordinate subspace maps to itsorthogonal complement which is a k -coordinate subspace as well.Now, let µ n,k : G n,k → ∆ n,k ⊂ R n be the standard moment map that is µ n,k ( L ) = 1 P | P J ( L ) | X | P J ( L ) | Λ J , where Λ J ∈ R n , Λ i = 1 for i ∈ J , while Λ i = 0 for i / ∈ J and J ⊂ { , . . . , n } , k J k = k .Assume that f ∈ Aut G n,k is an automorphism for which there exists a combina-torial isomorphism ¯ f : ∆ n,k → ∆ n,k such that the following diagram commutes: G n,k f −−−→ G n,k y µ n,k y µ n,k ∆ n,k ¯ f −−−→ ∆ n,k , (1)that is µ n,k ◦ f = ¯ f ◦ µ n,k . Lemma 2.
If an automorphism f of G n,k satisfies the assumption (1) then f ∈ T n − ⋊ S n for n = 2 k and f ∈ T n − × ( Z × S n ) for n = 2 k .Proof. Since µ n,k gives a bijection between the set W of fixed points for T n -action on G n,k and the vertices of ∆ n,k , and, by assumption ¯ f is a combinatorialisomorphism, it follows that f leaves the set W invariant. The statement followsfrom Proposition 1.We prove now that the automorphisms which satisfy (1) in fact coincide with T n − ⋊ S n or Z × ( T n − ⋊ S n ) . 7ince µ n,k is T n -invariant it follows that any f ∈ T n − < Aut G n,k satisfies (1) asone can take ¯ f = id ∆ n,k . The action of symmetric group S n on G n,k induces theaction of S n on ∆ n,k given by the permutations of coordinates. Lemma 3.
For any s ∈ S n the combinatorial automorphism ¯ s : ∆ n,k → ∆ n,k such the diagram (1) commutes is given by the permutation of coordinates in R n which is defined by s .Proof. Since P J ( s ( L )) = P s ( J ) ( L ) = P J ( L ) for any L ∈ G n,k it follows that µ n,k ( s ( L )) = (( pr ◦ µ n,k )( s ( L )) , . . . , ( pr n ◦ µ n,k )( s ( L )) = (( pr s (1) ◦ µ n,k )( L ) , . . . , ( pr s ( n ) ◦ µ n,k )( L )) = ¯ s ( µ n,k ( L )) , where pr i : R n → R i is the projection on i -th coordinate, ≤ i ≤ n .As for the involutive automorphism of G k,k we first recall an explicit descriptionof the diffeomorphism c n,k : G n,k → G n,n − k . Let L ∈ G n,k and assume thatthe Pl¨ucker coordinate P J ( L ) = 0 , where J = { , . . . , k } . Then there exists thebase for L in which L can be represented by n × k -matrix whose submatrix whichconsists of the first k columns is an identity matrix, that is L = (cid:18) IA (cid:19) , where A is ( n − k ) × k matrix . (2)It is easy to check that the ( n − k ) -dimensional subspace l − ( L ′ ) can be repre-sented by the matrix l − ( L ′ ) = (cid:18) − A T I (cid:19) , where I is ( n − k ) × ( n − k ) identity matrix . (3)It immediately implies the following: Lemma 4.
For any J ⊂ { , . . . , n } , k J k = k it holds P J ( L ) = ± P ¯ J ( l − ( L ′ )) , where ¯ J = { , . . . , n } \ J. (4)Then Lemma 4 implies: Lemma 5.
There exists a combinatorial isomorphism ¯ c n,k : ∆ n,k → ∆ n,n − k suchthat the following diagram commutes G n,k c n,k −−−→ G n,n − k y µ n,k y µ n,n − k ∆ n,k ¯ c n,k −−−→ ∆ n,n − k . roof. Consider an isomorphism ¯ c n,k : ∆ n,k → ∆ n,n − k which sends x = P J ⊂{ ,...,n } , k J k = k α J Λ J to ¯ c n,k ( x ) = P J ⊂{ ,...,n } , k J k = k α ¯ J Λ ¯ J , where ¯ J = { , . . . n } \ J . It directly followsfrom (4) that ¯ c n,k is a required combinatorial isomorphism.For n = 2 k we obtain the automorphism ¯ c k,k : ∆ k,k → ∆ k,k which is given by ¯ c k,k ( x ) = X J ⊂{ ,..., k } , k J k = k α ¯ J ( − Λ J ) , (5)where x = P J ⊂{ ,...,n } , k J k = k α J Λ J and = (1 , . . . , .Together with Lemma 5 this implies: Lemma 6.
For n = 2 k the isomorphism ¯ c k,k : ∆ k,k → ∆ k,k is given by x → − x, (6) where = (1 , . . . , .Proof. For x = ( x , . . . , x k ) ∈ ∆ k,k we have that x i = 1 P J, k J k = k | P J ( L ) | X J,i ∈ J, k J k = k | P J ( L ) | for some L ∈ G n,k . Then from (5) it follows that ¯ c k,k ( x ) i = 1 P J ⊂{ ,...,n } , k J k = k | P J ( L ) | X J ⊂{ ,....n }\{ i } , | J k = k | P J ( L ) | =1 k ( x + . . . + x n − ( k − x , . . . , x + . . . + x n − − ( k − x n ) = (1 − x , . . . , − x n ) , since x + . . . x n = k .Altogether we obtain: Proposition 2.
An element f ∈ G n,k satisfies the assumption (1) if and only if f ∈ T n − ⋊ S n for n = 2 k and f ∈ Z × ( T n − ⋊ S n ) for n = 2 k . ∆ n,k by the moment map it holds: Lemma 7. If f ∈ Aut G n,k satisfies assumption (1) then µ − n,k ( x ) is homeomorphicto µ − n,k ( ¯ f ( x )) for any x ∈ ∆ n,k .Proof. From (1) it directly follows that f : µ − n,k ( x ) → µ − n,k ( ¯ f ( x )) . Since f is anautomorphism the statement follows.Since any automorphism for G n,k which commutes with T n -action produces ahomeomorphism of G n,k /T n and the moment map is T n -invariant, we deducefrom Proposition 2 the following. Corollary 1.
For n = 2 k , the subspaces µ − n,k ( x ) /T n , µ − n,k ( s ( x )) /T n ⊂ G n,k /T n are homeomorphic for any x ∈ ∆ n,k and any s ∈ S n . For n = 2 k , the subspaces µ − n,k ( x ) /T n , µ − n,k ( ¯ f ( x )) /T n are homeomorphic for any x ∈ ∆ n,k and any ¯ f ∈ S n or ¯ f = − x . G n, We first recall the notions of an admissible polytope and a stratum as introducedin [7]. Some other equivalent definition may be found in [11]. Let M ij = { L ∈ G n, | P ij ( L ) = 0 } , where { i, j } ⊂ { , . . . , n } , i < j be the standard Pl¨uckercharts on G n, and put Y ij = G n,k \ M ij . A nonempty set W σ = (cid:0) \ { i,j }∈ σ M ij (cid:1) ∩ (cid:0) \ { i,j } / ∈ σ Y ij (cid:1) is called a stratum, where σ ⊂ (cid:0) n (cid:1) . For a stratum W σ we have that µ ( W σ ) = ◦ P σ ,where P σ is a convex hull of the vertices Λ ij , { i, j } ∈ σ . A polytope P σ whichcan be obtained in this way is said to be an admissible polytope. In addition, allpoints from W σ has the same stabilizer T σ ⊆ T n and the torus T σ = T n /T σ actsfreely on W σ , see [7].The boundary ∂ ∆ n, consists of n copies of the hypersimplex ∆ n − , and n copiesof the simplex ∆ n − , . Moreover, µ − ( ∂ ∆ n, ) = n µ − (∆ n − , ) ∪ n µ − (∆ n − , ) = G n − , ∪ n C P n − . It follows that the admissible polytopes for G n, can bedescribed inductively, by describing those which intersect the interior of ∆ n, . Wehave that dim ∆ n, = n − and that dim P σ is equal to the dimension of the torus T σ which acts freely on the stratum W σ for any admissible polytope P σ in ∆ n, ,see [7].Note that one can find in [13] as well the description of the admissible polytopes P σ ⊂ ∆ n, for T n -action in G n, in terms of matroid theory and Chow quotient G n, // ( C ∗ ) n : there are first described all matroid decomposition of ∆ n, and after-wards it is proved that there are all realizable, that is all of them come from somenon-empty Chow strata in G n, // ( C ∗ ) n .We provide here the new purely analytical description of the admissible polytopesfor G n, , which is going to be suitable for our purpose of the description of anorbit space G n, /T n .We find useful to point to the following observation. Let pr i : R n → R i denotesthe projection on i -th coordinate, ≤ i ≤ n . Lemma 8.
A point x ∈ ∆ n,k belongs to the boundary ∂ ∆ n,k if and only if pr i ( x ) =0 or pr i ( x ) = 1 for some ≤ i ≤ n . Moreover, a polytope P which is the convexspan of some vertices of ∆ n,k belongs to ∂ ∆ n,k if and only if there exists ≤ i ≤ n such that pr i ( x ) = 0 for all x ∈ P or pr i ( x ) = 1 for all x ∈ P . Proposition 3.
Any admissible polytope for G n, whose dimension is ≤ n − belongs to the boundary ∂ ∆ n, for ∆ n, .Proof. Let P σ be an admissible polytope and let dim P σ = q ≤ n − . Because ofthe action of the symmetric group, we can assume that the V = (1 , , , . . . , is avertex for P σ . Then there exist q vertices V , . . . , V q for P which are adjacent to V and the dimension of the subspace G spanned by the vectors V − V , . . . , V − V q is equal to q . The polytope P σ is the intersection of hypersimplex ∆ n, and the q -dimensional plane which contains the point V and which is directed by thesubspace G . Since V , . . . V q are adjacent to V they must have and at the firsttwo coordinate places. Thus, any of V i ’s, ≤ i ≤ q has exactly one and thelast n − places. Since q ≤ n − , all V i , ≤ i ≤ q must have one commoncoordinate x j , j ≥ which is equal to zero. It implies that P σ belongs to theboundary ∂ ∆ n, . 11 .2 Admissible polytopes of dimension n − The admissible polytopes of dimension n − can be divided into those whichbelong to the boundary ∂ ∆ n, and those which intersect the interior of ∆ n, . Thoseon ∂ ∆ n, are given by n hypersimplices ∆ n − , and their ( n − -dimensionaladmissible polytopes, and ∆ n − , which is the simplex ∆ n − .We describe here those admissible ( n − -dimensional polytopes which intersectthe interior of ∆ n, .Denote by Π { i,j } the set of all ( n − -dimensional planes which contain the vertex Λ { i,j } = e i + e j , which are parallel to the edges of ∆ n, that are adjacent to Λ { i,j } ,and which intersect the interior of ∆ n, . The edges adjacent Λ { i,j } are given by e { j,s } = Λ { i,j } − Λ { i,s } , s = i, j,e { i,q } = Λ { i,j } − Λ { q,j } , q = i, j. Note that the vectors e { j,m } and e { i,s } represent the complementary roots for U (2) × U ( n − related to U ( n ) and G n, can be represented as the homogeneous space U ( n ) /U (2) × U ( n − . The planes Π { i,j } can be described as follows: Lemma 9.
The set Π { i,j } consists of the planes α S { i,j } ,l = Λ { i,j } + F l,S , ≤ l ≤ n − , S ⊂ { , . . . , n } , k S k = l and i, j / ∈ S . The directrix F l,S is spanned by the vectors e { j,s } and e { i,q } , s ∈ S, q / ∈ S ∪ { i, j } . It is easy to verify that the planes α S { i, } ,l can be more explicitly written: Corollary 2.
The set Π { i,j } consists of the ( n − -dimensional planes which areobtained as the intersection of the plane n P i =1 x i = 2 with the planes x j + X s ∈ S x s = 1 , (7) where S ⊂ { , . . . , n } , k S k = l , ≤ l ≤ n − and i, j / ∈ S . These planescoincide with the planes x i + X s / ∈{ i,j }∪ S x s = 1 . (8)12he group S n acts on the set { Π { i,j } , ≤ i < j ≤ n } . The stabilizer of S n -actionon the set { Π { i,j } , ≤ i < j ≤ n } is the group S × S n − , that is the group S × S n − acts on the set Π { i,j } .We first prove the following: Proposition 4.
The admissible polytopes of dimension n − which do not belongto the boundary ∂ ∆ n, coincide with the polytopes obtained by the intersection of ∆ n, with the planes from Π { i,j } , where ≤ i < j ≤ n .Proof. Let P σ be an admissible polytope which does not belong to the boundary ∂ ∆ n, . Because of the action of the symmetric group S n , we can assume that Λ { , } is a vertex of P σ . Since dim P σ = n − it follows that Λ { , } has n − lin-early independent adjacent vertices in P σ . Any of these vertices has one and one at the first two coordinate places. Let l be the number of these vertices having as the first coordinate, then they have exactly one at the last n − coordinates.Due to the S n -action we can assume that these vertices are Λ { , } , . . . , Λ { ,l +2 } .The other n − − l vertices have as the first coordinate, as the second coordi-nate and they all have exactly one at the last n − coordinates. We claim that has to be among the last n − l − coordinates for all these n − − l vertices. Ifthis is not the case, one of these vertices would have the last n − − l coordinatesall equal to zero. For the other of these vertices we eventually would have at thelast n − l − coordinate places, and these places have to be different for differentvertices. The number of such vertices is at most n − − l , so they must have onecommon of the last n − l − coordinates which is equal to zero and consequentlyit is zero coordinate for all n − − l vertices, that is for all n − vertices. It wouldimply that the polytope P σ belongs to ∂ ∆ n, .Therefore, the other n − − l vertices are Λ { ,j } , l + 2 ≤ j ≤ n . It follows thatthe polytope P σ , up to the action of the symmetric group S n belongs to the plane α { , } ,l = α S { , } ,l where S = { , . . . , l + 2 } for some ≤ l ≤ n − .Let now P be a polytope, which is obtained as the intersection of ∆ n, with aplane from the set Π { , } , up to the action of the group S n . The points of the plane α { , } ,l can be explicitly written in R n as: (1 + a l +1 + . . . + a n − , a + . . . + a l , − a , . . . , − a n − ) , a i ∈ R , ≤ i ≤ n − . It follows that the vertices of ∆ n, which belong to this plane are Λ { ,j } , ≤ j ≤ l + 2 , Λ { , } j , l + 3 ≤ j ≤ n and Λ { i,j } , ≤ i ≤ l + 2 , l + 3 ≤ j ≤ n , that is P isthe convex hull of these vertices. If we consider the point L ∈ G n, given by the13atrix A L such that a = a = 1 , a = a = 0 , a i = a j = 0 , ≤ i ≤ l + 2 , l + 3 ≤ j ≤ n and a j = 0 , l + 3 ≤ j ≤ n , a j = 0 , ≤ j ≤ l + 2 , we seethat the image by the moment map of the closure ( C ∗ ) n -orbit of the point L is thepolytope P . Thus, P is an admissible polytope. Proposition 5.
The number of irreducible representations for S × S n − -actionon Π { i,j } is [ n − ] . The dimensions of these irreducible representations are:for n odd : (cid:18) n − l (cid:19) , ≤ l ≤ [ n −
22 ] , for n even : (cid:18) n − l (cid:19) , ≤ l < [ n −
22 ] and n − (cid:18) n − n − (cid:19) Proof.
The number of planes in the set Π { i,j } is | Π { i,j } | = n − X l =1 (cid:18) n − l (cid:19) = 2 n − − . It implies that the group S n − acts on the set Γ { i,j } = Π { i,j } /S which consists of q = 2 n − − elements. Moreover, from the description of the planes from Π { i,j } itfollows that the set of generators for S n − - action on Γ { i,j } is given by the planes α { i,j } ,l = α S { i,j } ,l , where S = { , . . . , l + δ { i,j } } , δ { i,j } = 0 , , corresponding tothe cases l < i < j , i ≤ l < j , i < j ≤ l and ≤ l ≤ [ n − ] . The stabilizer ofthe element α { i,j } ,l is S l × S n − − l for ≤ l < [ n − ] . For l = [ n − ] and n odd,that is l = n − , the stabilizer is S l × S n − − l , while for n even, that is l = n − , thestabilizer is S l × S l × S l . It follows that the S n − action on Γ { i,j } corresponds tothe representation of S n − in C n − − whose irreducible summands for n odd arein dimensions ( n − l !( n − − l )! , ≤ l ≤ [ n − ] , while for n even they are in dimensions ( n − l !( n − − l )! , ≤ l < [ n − ] and ( n − n − !) .In this way, the Proposition 4 can be improved as follows: Corollary 3.
The admissible polytopes of dimension n − which do not belongto the boundary ∂ ∆ n, are , up to the action of the symmetric group S n , given bythe intersection of ∆ n, with the planes α { , } ,l , where ≤ l ≤ [ n − ] . Using Corollary 2 we can summarize the previous result as follows:14 heorem 1.
The admissible polytopes for G n, of dimension n − which havenon-empty intersection with ◦ ∆ n, are given by the intersection of ∆ n, with the set Π which consists of the planes: X i ∈ S, k S k = p x i = 1 , where S ⊂ { , . . . , n } , ≤ p ≤ [ n . (9)We also find useful to note the following: Corollary 4.
An admissible polytope of dimension n − defined by the hyperplane x k + P i ∈ S,k / ∈ S x i = 1 , is a convex span of the vertices Λ i,k for i ∈ S , where S ⊂{ , . . . , n } , k S k = p , ≤ p ≤ [ n ] and ≤ k ≤ n . We describe as well the numbers of vertices of these polytopes:
Corollary 5.
An admissible polytope of dimension n − which does not belongto the boundary of ∆ n, has n p = p ( n − p ) , vertices for some ≤ p ≤ [ n ] .Moreover, the number q p of admissible polytopes which have n p , ≤ p ≤ [ n ] vertices is q p = (cid:18) np (cid:19) for n odd ,q p = (cid:18) np (cid:19) for n even and ≤ l < [ n −
22 ] ,q n = (cid:0) n n (cid:1) for n even . Example 1.
It follows from Corollary 3 that G , has one generating admissiblepolytope of dimension which is inside ∆ , . This polytope has vertices and its S -orbit consists of polytopes, which give all admissible polytopes of dimension that are in ◦ ∆ , . These polytopes are defined by the planes x + x = 1 , x + x =1 and x + x = 1 . Example 2.
The Grassmannian G , has one generating admissible interior poly-tope of dimension which is inside ∆ , . It has vertices and its S -orbit has elements which give all admissible polytopes of dimension that are in ◦ ∆ , .These polytopes are defined by the planes x i + x j = 1 where ≤ i < j ≤ . xample 3. The Grassmannian G , has generating admissible polytopes ofdimension inside ∆ , . These polytopes have and vertices and their S -orbits consists of and elements respectively. They are defined by the planes x i + x j = 1 , ≤ i < j ≤ and x + x j + x k = 1 , ≤ i < j ≤ . Note thathere the corresponding representation for S × S - action on C has irreduciblesummands of dimension and . n − Before to describe the admissible polytopes of dimension n − , we first need thefollowing result. Lemma 10.
Assume that the points of an admissible polytope P σ , dim P σ = n − satisfy inequalities X i ∈ I x i ≤ and X j ∈ J x j ≤ , where I, J ⊂ { , . . . , n } . If I ∩ J = ∅ then the points of P σ satisfy as well aninequality X s ∈ I ∪ J x s ≤ . Proof. If I ∩ J = ∅ then P σ does not contain the vertices Λ ij , i ∈ I , j ∈ J , that is P ij ( L ) = 0 for all points L from the stratum W σ . Now if x = ( x , . . . , x n ) ∈ P σ then x = µ ( L ) for some L ∈ W σ . Note that for s ∈ I ∪ J we have x s = S s S , S s = X m/ ∈ I ∪ J P sm ( L ) , and S = P ≤ p The admissible polytopes for ∆ n, of dimension n − are given by ∆ n, and the intersections with ∆ n, of all collections of the half-spaces of theform H S : X i ∈ S x i ≤ , S ⊂ { , . . . , n } , k S k = k, ≤ k ≤ n − , such that if H S , H S belongs to a collection then S ∩ S = ∅ . It follows that any polytope which can be obtained as the intersection with ∆ n, of a half-space of the form P s ∈ S x s ≤ , where S ⊂ { , . . . , n } , k S k = k and ≤ k ≤ n − is an admissible ( n − -dimensional polytope. Also, any polytopewhich can be obtained as the intersection with ∆ n, of the intersection of half-spaces of the form P i ∈ I x i ≤ and P j ∈ J x j ≤ such that I ∩ J = ∅ is an admissiblepolytope, where k I k , k J k ≥ . Continuing in this way we describe all admissiblepolytopes of dimension n − . Proof. Any admissible polytope P σ of dimension n − which is not ∆ n, has afacet which intersects ◦ ∆ n, .1) If P σ has exactly one facet which intersect ◦ ∆ n, then according to Theorem 1,we have that P σ is given by those points in ∆ n, which satisfy one of the inequal-ities P i ∈ I x i ≤ or P i ∈ I x i ≥ for some I ⊂ { , . . . , n } , k I k = l and ≤ l ≤ [ n ] .Since the points from ∆ n, satisfy n P i =1 x i = 2 it follows that the second inequalityis equivalent to P j ∈{ ,...,n }\ I x j ≤ .2) Assume that P σ has two facets which intersect ◦ ∆ n, and which are given by X i ∈ I x i = 1 and X j ∈ J x j = 1 , (10)17hich is equivalent to X i ∈{ ,...,n }\ I x i = 1 and X j ∈{ ,...,n }\ J x j = 1 , (11)where I, J ⊂ { , . . . , n } , k I k , k J k ≥ . Therefore, we can always assume that P σ is given by X i ∈ I x i ≤ and X j ∈ J x j ≤ , (12)Then Lemma 10 implies that I ∩ J = ∅ .Arguing by induction one can prove in the same way the statement for an admis-sible polytope P σ with an arbitrary finite number of interior facets. Example 4. It follows from Theorem 2 that the admissible polytopes for G , ofdimension are ∆ , and the intersection of ∆ , with the half spaces x i + x j ≤ , ≤ i < l ≤ . There are (cid:0) (cid:1) = 6 such polytopes. Example 5. The admissible polytopes for G , of dimension are: ∆ , and thepolytope obtained by the intersection of ∆ , with the1. half spaces x i + x j ≤ , ≤ i < j ≤ 2. half space x i + x j + x k ≤ , ≤ i < j < k ≤ , which can be written as x p + x q ≥ , ≤ p ≤ q ≤ 3. intersection of the half spaces x i + x j ≤ and x p + x q ≤ , where { i, j } ∩{ p, q } = ∅ , ≤ i < j ≤ , ≤ p < q ≤ .There are polytopes of type (1) and any of them has vertices, there are polytopes of type (2) and any of them has vertices and there are polytopes oftype (3) and any of them has vertices. Example 6. The admissible polytopes for G , of dimension are: ∆ , and thepolytope obtained by the intersection of ∆ , with the1. half spaces x i + x j ≤ , ≤ i < j ≤ ;2. half spaces x i + x j + x k ≤ , ≤ i < j < k ≤ ;3. half spaces x i + x j + x k + x l ≤ , ≤ i < j < k < l ≤ which can bewritten as x p + x q ≥ , ≤ p < q ≤ ; . intersection of the half spaces x i + x j ≤ and x p + x q ≤ , where { i, j } ∩{ p, q } = ∅ , ≤ i < j ≤ , ≤ p < q ≤ .5. intersection of the half spaces x i + x j ≤ and x p + x q + x s ≤ , where { i, j } ∩ { p, q, s } = ∅ , ≤ i < j ≤ , ≤ p < q < s ≤ .The number of these polytopes and their vertices are as follows: type (1) - (15 , ,type (2) - (20 , , type (3) - (15 , , type (4) - (45 , and type (5) - (60 , . G n, The algebraic torus ( C ∗ ) n acts canonically on G n, and its action is an extension of T n - action. The strata W σ are invariant under ( C ∗ ) n -action and the correspondingorbit spaces F σ = W σ / ( C ∗ ) n we call the spaces of parameters of the strata. In thissection we discuss the spaces of parameters of the strata in G n, whose admissiblepolytopes intersect the interior of ∆ n, . Moreover, we show that the space F n described in [9] is a universal space of parameters for G n, and describe the virtualspaces of parameters of these strata. The notions of universal space of parametersan virtual spaces of parameters are defined in [7] and [8]. G n, Let us fix the chart M , this can be done without loss of generality because of theaction of symmetric group S n . The elements of this chart can ne represented bythe matrices which have the first × -minor equal to identity matrix. We con-sider the coordinates for the points from M as the columns of the correspondingmatrices that is ( z , . . . , z n , w , . . . , w n ) . The charts are invariant for the action ofthe algebraic torus ( C ∗ ) n and this action is, in the local coordinates of the chart M , given by ( t t z , . . . , t n t z n , t t w , . . . , w n t w n ) . (13)If we put τ = t t , . . . , τ n − = t n t , τ n − = t t , we obtain that t i t = τ i − · τ n − τ , ≤ i ≤ n. .1.1 The main stratum The main stratum W is characterized by the condition that its points have non-zero Pl¨ucker coordinates, it belongs to any chart and its admissible polytope is ∆ n, . The main stratum can be, in the chart M , written by the system of equations c ′ ij z i w j = c ij z i w j , ≤ i < j ≤ n, (14)where the parameters ( c ′ ij : c ij ) ∈ C P and c ij , c ′ ij = 0 and c ij = c ′ ij for all ≤ i < j ≤ n .The number of parameters is N = (cid:0) n − (cid:1) and it follows from (14) that these pa-rameters satisfy the following equations c ′ i c j c ′ ij = c i c ′ j c ij , ≤ i < j ≤ n. (15)The number of these equations is M = (cid:0) n − (cid:1) . . From (15) we obtain that ( c ij : c ′ ij ) = ( c ′ i c j : c i c ′ j ) , ≤ i < j ≤ n. (16) ( c i : c ′ i ) = ( c j : c ′ j ) , ≤ i < j ≤ n. (17)Note that (15) gives the embedding of the space F n = W/ ( C ∗ ) n in ( C P ) N , N = (cid:0) n − (cid:1) . Moreover, it follows from (15) that F n is an open algebraic manifoldin ( C P ) N given by the intersection of the cubic hypersurfaces (15) and the con-ditions that ( c ′ ij : c ij ) ∈ C P \ A . The dimension of F n is n − what is exactlyequal to N − M ) . Any stratum W σ consists of those points from G n, for which some fixed Pl¨uckercoordinates are zero. If W σ ⊂ M then the stratum W σ is defined by the condition P i = 0 , P j = 0 and P pq = 0 , for some ≤ i, j ≤ n , ≤ p < q ≤ n . Inthe local coordinates of the chart M this condition translates to w i = z j = 0 and z p w q = z q w p . Therefore, according to Lemma 14.10 from [8] any stratum W σ ⊂ M is obtained by restricting the surfaces (14) to some C J , where J ⊂{ (3 , , (3 , , . . . , ( n − , n ) , ( n, n ) } and k J k = l for some ≤ l ≤ Q , where Q = ( n − . In particular, if an admissible polytope for W σ has a maximal20imension n − we have that l ≥ n − . It implies that if the space of parameters F σ = W σ / ( C ∗ ) n for W σ is not a point, then it can be obtained by restricting theintersection of the cubic hypersurfaces (15) to some q factors C P B = C P \ B in ( C P ) N , where B = { (1 : 0) , (0 : 1) } and ≤ q ≤ l . Proposition 6. If the admissible polytope P σ of a stratum W σ intersects the inte-rior of ∆ n, and dim P σ = n − then the space of parameters F σ is a point.Proof. Since P σ is an interior polytope in ∆ n, it follows from Lemma 8 thatthere must exist z i , w j = 0 for some ≤ i, j ≤ n . Because of the action of thesymmetric group we can assume that z i = 0 for ≤ i ≤ l ≤ n . Note that l is ≥ , but it must be that l ≤ n − . This is because dim P σ = n − , so thestabilizer for T n action on W σ is of dimension , that is T n − is a maximal toruswhich acts freely on W σ . Then we have that w j = 0 for l + 1 ≤ j ≤ n and w j = 0 for ≤ j ≤ l . It implies that in the chart M the stratum W σ is given by thecoordinates ( z , . . . z l , , . . . , , w l +1 , . . . , w n ) , which means that W σ consists ofone ( C ∗ ) n -orbit, that is F σ is a point. G n, The universal space of parameters F n for a (2 n, k ) - manifold M n with an effec-tive action of the compact torus T k , k ≤ n , is a compactification of the space ofparameters F n of the main stratum, see [8]. The universal space of parametersfor the Grassmannian G , regarding to the canonical action of T is defined andexplicitly described in [7]. The general notion of universal space of parametersis axiomatized in [8]. In [19] it is proved that the Chow quotient G n, // ( C ∗ ) n asdefined in [13] provides an universal space of parameters for G n, regarded to thecanonical T n -action. The method in [19] is based on the embedding of the space F n in C P N , N = (cid:0) n +14 (cid:1) via the cross ratio of the Pl¨ucker coordinates. In thispaper we show that the space F n described in [9] is a universal space of parame-ters for T n -action on G n, . This space F n is obtained in [9] by the techniques ofa wonderful compactification of the space of parameters F n of the main stratum,starting form the requirement that the homeomorphisms of F n which are definedby the transition functions between the standard Pl¨ucker charts for G n, extendsto the homeomorphisms for F n .Let us fix the chart M and consider a stratum W σ ⊂ M . As already remarked,in the local coordinates for the chart M this stratum is defined by the conditions21 s = w m = 0 and z i w j = z j w i for some ≤ s, m ≤ n and ≤ i < j ≤ n . Onthe other hand the main stratum W is in the chart M given by (15) and it is adense set in G n, . Using these two facts, one can try to assign the new space ofparameters ˜ F σ to W σ . The question which arises here is in which ambient spacethat is a compactification F for F the corresponding assignment is to be done.The answer to this question is determined by the condition that such assignmentmust not depend on a fixed chart.First it is obvious that F n has to contain the closure ¯ F n of F n in ( C P ) N . Thisclosure is given by the intersection (15) in ( C P ) N . Second, the transition func-tions between the charts produce the homeomorphisms between the records of thespace F n in these charts. The compactification F n should be such that these home-omorphisms extend to the homeomorphisms of F n for an arbitrary two charts.Starting from these requirements the following theorem is proved in [9]: Theorem 3. Let the manifold F n is obtained by the wonderful compactification of ¯ F n with the generating set of subvarieties ¯ F I ⊂ ¯ F n , defined by ( c ik : c ′ ik ) = ( c il : c ′ il ) = ( c kl : c ′ kl ) = (1 : 1) for ikl ∈ I , where I ⊂ { ikl | ≤ i < k < l ≤ n } .Then any homeomorphism f ij,kl : F n → F n induced by the transition functionbetween the charts M ij and M kl extends to the homeomorphism of F n . We can now proceed as in the case of G , . To any stratum W σ we assign thevirtual space of parameters ˜ F σ, ⊂ F n in the chart M using the fact that themain stratum W is a dense set in G n, . In this regard we differentiate the followingcases.1. W σ ⊂ M - we assign ˜ F σ, ⊂ F n using the description (14) of the mainstratum.2. W σ ∩ M = ∅ - we consider a chart M ij such that W σ ⊂ M ij and assignto W σ the space ˜ F σ,ij using (14). Then using homeomorphism for F n statedby Theorem 3 we assign to W σ the subset ˜ F σ, ⊂ F n which is the imageof ˜ F σ,ij F n by this homeomorphism. Proposition 9.11 from [7] applies heredirectly to show that ˜ F σ, does not depend on the choice of a chart M ij which contains W σ . Remark . By the action of the symmetric group S n the same construction holdsfor an arbitrary chart M ij , that is to any stratum W σ one can assign ˜ F σ,ij ⊂ F n .22 emark . It follows from (14) that for the main stratum it holds ˜ F ij ∼ = F .Moreover, if the stratum W σ is defined by the condition that is has exactly onezero Pl¨ucker coordinate it also follows from (14) that ˜ F σ,ij ∼ = F σ . This is alsotrue for the strata which have exactly two zero Pl¨ucker coordinates P ij and P kl where i, j = k, l . For all other strata it follows from (14) that the correspondingvirtual and real spaces of parameters are not homeomorphic, that is the virtualspaces of parameters are ”bigger”. For example, the space of parameters of thestratum in G n, defined by P = P = P = 0 is, by the discussion pre-ceding Proposition 6 homeomorphic to ( C P A ) ( n − n − subject to the relationsof the type (15). Its universal space of parameters is by (14) homeomorphic to C P × (1 : 0) n − × ( C P A ) ( n − n − subject to the relations (15).It si straightforward to verify that for an arbitrary chart M ij it holds [ σ ˜ F σ,ij = F n . (18) Remark . To illustrate (18) we follows [7] and consider the Grassmann manifold G , and fix the chart M . In the procedure of the wonderful compactification F is obtained by the blowing up of ¯ F at the point S = ((1 : 1) , (1 : 1) , (1 : 1)) .Now, in the chart M the space of parameters of the stratum W σ defined by P = P = P = 0 and P ij = 0 for all others i, j , is the point S . The virtual space ofparameters for W σ we can explicitly obtain if we look at this stratum in the chart M . Namely, the local coordinates for W σ in M are z , z = z = 0 , w , w , w .Therefore, it follows from (14) that ˜ F σ, = (1 : 0) × (1 : 0) × C P ∼ = C P , whichimplies that ˜ F σ, ∼ = C P , that is ˜ F σ, is homeomorphic to the exceptional divisorat the point S .In addition, in an analogous way as it was done for G , in [7], paragraph 9.2, itcan be proved: Proposition 7. There exists a canonical projection g σ,ij : ˜ F σ,ij → F σ for anyadmissible set σ and any chart M ij .Remark . From now on we will omit in the notation of the virtual spaces ofparameters the indices of the charts and write jut ˜ F σ .Altogether, the proof of Theorem 11.11 for G , directly generalizes to G n, for n ≥ , which verifies that the third condition of Axiom 6 for (2 n, k ) -manifoldsfrom [8] is satisfied. In this way we obtain:23 heorem 4. The space F n is the universal space of parameters for G n, . Example 7. For n = 5 the universal space of parameters F is obtained as theblow up of the smooth algebraic manifold ¯ F = { (( c : c ′ ) , ( c : c ′ ) , ( c : c ′ )) ∈ ( C P ) | c ′ c c ′ = c c ′ c } at the point ((1 : 1) , (1 : 1) , (1 : 1)) . Thisresult is obtained in [7], see also [9] for more general insight. Example 8. For n = 6 the universal space of parameters F is obtained in [9] bythe wonderful compactification of the algebraic manifold ¯ F ⊂∈ ( C P ) definedby c ′ c c ′ = c c ′ c , c ′ c c ′ = c c ′ c ,c ′ c c ′ = c c ′ c , c ′ c c ′ = c c ′ c , with generating subvarieties ¯ F = ¯ F ∩ { ( c : c ′ ) = ( c : c ′ ) = ( c : c ′ ) = (1 : 1) } ¯ F = ¯ F ∩ { ( c : c ′ ) = ( c : c ′ ) = ( c : c ′ ) = (1 : 1) } ¯ F = ¯ F ∩ { ( c : c ′ ) = ( c : c ′ ) = ( c : c ′ ) = (1 : 1)) } ¯ F = ¯ F ∩ { ( c : c ′ ) = ( c : c ′ ) = ( c : c ′ ) = (1 : 1) } . G n, /T n The notion of a critical point on G n, /T n can be defined in one of the equivalentways. Consider a moment map µ n, : G n, → ∆ n, . It is a smooth map and thereare, in the standard way, defined the critical points and the critical values of thismap. It is proved in [8] that a point L ∈ G n, is a critical point of the moment map µ n, if and only if the stabilizer of L related to the standard T n -action on G n, isnon-trivial. This is equivalent to say that the admissible polytope of the stratumwhich contains L is not of maximal dimension n − .We say that a point [ L ] ∈ G n, /T n is a critical point if L ∈ G n, is a criticalpoint in the above sense. This is correctly defined since obviously the notion of acritical point on G n, is invariant for T n -action.The notion of the critical points in G n, /T n can be as well related to the singu-larities of G n, /T n following tubular neighborhood theorem. Precisely, the tubu-lar neighborhood theorem states that for any point L ∈ G n, there exists T n -equivariant diffeomorphism between the vector bundle T n × T L V and the neigh-borhood of the orbit T n · L in G n, , where T L is the stabilizer of the point L and V 24s the normal bundle in ( T G n, ) T n · L to the tangent bundle T ( T n · L ) . It implies thatthere exists a neighborhood in the orbit space G n, /T n of the point defined by theorbit T n · L which has the form ( T n × T L V ) /T n = V T L × cone ( S ( U ) /T L ) , where V T L is the subspace of V consisting of the vectors fixed by T L , U is subspace of V defined by V = V T L ⊕ U related to some T n - invariant metric on G n, and S ( U ) is the corresponding unit sphere. In this way one concludes that any pointin G n, /T n which is defined by a point from G n, having the non-trivial stabilizer,has a neighborhood with cone-like singularities. In this way all singularities ofthe orbit space G , /T ∼ = S are described in [6].On the other hand to any stratum W σ ⊂ G n, we assigned the corresponding spaceof parameters F σ and the virtual space of parameters ˜ F σ . As we noted in Remark 2these spaces are in general not homeomorphic. We say that a point L ∈ G n, isa singular point for T n -action on G n, if the space of parameters of the stratum W σ such that L ∈ W σ is not homeomorphic to the virtual space of parameters for W σ . Since the notions of spaces of parameters and virtual spaces of parametersare obviously invariant for the standard T n -action we can define the notion of asingular point in the orbit space G n, /T n . Definition 1. A point [ L ] = T n · L ∈ G n, /T n is said to be a singular point forthe standard T n -action on G n, if the space of parameters F σ of the stratum W σ , L ∈ W σ is not homeomorphic to the virtual space parameters ˜ F σ for W σ . The singular points can be characterized in terms of the Pl¨ucker coordinates asfollows. Proposition 8. A point [ L ] ∈ G n, /T n is a singular point if and only if thereexists i , ≤ i ≤ n such that P ij ( L ) = 0 for all j = i , ≤ j ≤ n or there exist ≤ i < j < k ≤ n such that P ij ( L ) = P ik ( L ) = P jk ( L ) = 0 .Proof. We first note that all points from from T n · L ⊂ G n, have the same Pl¨uckercoordinates. Let [ L ] is a singular point in G n, /T n and let L ∈ W σ . If P σ ⊂ ∂ ∆ n, then by Lemma 8 we have that P σ belongs to the plane x i = 0 or to the plane x i = 1 for some ≤ i ≤ n . It implies that P ij ( L ) = 0 for all j = i or P jk ( L ) = 0 for all j, k = i . Thus, the point L satisfies the condition of the statement. If P σ ∩ ◦ ∆ n, = ∅ , assume that the vertex Λ ∈ P σ . Then W σ belongs to the chart M and let z , . . . , z n , w , . . . , w n be the local coordinates in this chart. Since F σ is not homeomorphic to ˜ F σ and P σ ∂ ∆ n, there exist ≤ i < j ≤ n such that z i = z j = 0 or w i = w j = 0 . It implies that P i ( L ) = P j ( l ) = P ij ( L ) = 0 or P i ( L ) = P j ( L ) = P ij ( L ) = 0 , that is the statement holds.25n proving opposite direction we can assume as well that W σ ⊂ M and that i =1 . Then j, k ≥ and in the local coordinates for M we have that w j = w k = 0 .Using (14) it implies that F σ and ˜ F σ are not homeomorphic, they differ at least by C P . Proposition 9. All critical points for G n, /T n are for n ≥ the singular pointsfor G n, /T n .Proof. Let [ L ] be a critical point in G n, /T n , assume that L belongs to the chart M and let z , . . . , z n , w , . . . , w n be the local coordinates in this chart. Sincethe stabilizer for L is non-trivial and n ≥ , it follows from (13) that there mustexist ≤ i < j ≤ n such that z i = z j = 0 or w i = w j = 0 or there must exist ≤ i ≤ n such that z i = w i = 0 . In other words there exist ≤ i < j ≤ n suchthat P i ( L ) = P j ( L ) = P ij ( L ) = 0 or P i ( L ) = P j ( L ) = P ij ( L ) = 0 or thereexists ≤ i ≤ n such that P ij ( L ) = 0 for any j = i . Then Proposition 8 impliesthat the point [ L ] is a singular point in G n, /T n . Remark . Proposition 9 does not hold for n = 4 . In that case the points which inthat chart M have coordinates z , z = 0 , w = 0 , w or z = 0 , z , w , w = 0 represent the points from G , /T which are critical, but which are not singularpoints. The points which have the local coordinates of this form in some chartexhaust all critical points in G , /T which are not singular.Let Sing X n denotes the set of singular point in X n and put Y n = X n \ Sing X n . Proposition 10. The set Y n ⊂ X n is a open, dense set in X n , which is a manifold.Proof. The boundary ∂W σ = W σ \ W σ is by [11] the union of the strata W ′ σ which correspond to the faces P σ ′ of the admissible polytope P σ , for any stratum W σ . It implies by 4.1.2 that F σ ′ ⊆ F σ and by 4.1.1 that ˜ F σ ⊆ ˜ F σ ′ , for any such σ ′ . Therefore, if a stratum W σ consists of singular points then W σ consists ofsingular points as well. It implies that Sing X n = ∪ W σ , where the union goesover all the strata consisting of singular points, so X n is a closed set. Thus, Y n isan open, dense set in X n , which is a manifold, as it contains the orbit space of themain stratum and it does not contain the critical points.26 The virtual spaces of parameters for G n, We establish now some properties of the virtual space of parameters for G n, which turn out to be important for the description of the orbit space G n, /T n .Let W σ be a stratum such that it admissible polytope satisfies P σ ∩ ◦ ∆ n, = ∅ . Then dim P σ = n − or dim P σ = n − . If dim P σ = n − it is proved in the previoussections that W σ is an one-orbit stratum while for dim P σ = n − it is not ingeneral. Theorem 5. If P σ is an admissible polytope such that dim P σ = n − and P σ ′ isa facet of P σ such that P σ ′ ∩ ◦ ∆ n, = ∅ then ˜ F σ ⊆ ˜ F σ ′ , where ˜ F σ and ˜ F σ ′ are the virtual spaces of parameters for W σ and W σ ′ .Proof. We first note that W σ ′ ⊂ W σ since W σ ′ is an one-orbit stratum. Assumewithout loss of generality that Λ is a common vertex for P σ and P σ ′ . Let P σ ′ begiven by the intersection of x i + . . . + x i l = 1 with ∆ n, , where ≤ i < i . . .
Let P σ ′ be an admissible polytope such that dim P σ ′ = n − and P σ ′ ∩ ◦ ∆ n, = ∅ . Then ˜ F σ ′ = [ σ ˜ F σ , (19) where the union goes over all σ such that P σ ′ is a facet of P σ . In an analogous way the same statement can be proved for an admissible polytope P σ ′ such that dim P σ ′ = n − and P σ ′ ⊂ ∂ ∆ n, , as any such polytope is definedby an additional condition x i = 0 or x i = 1 for some ≤ i ≤ n . Altogether wehave: Proposition 11. Let P σ ′ be an admissible polytope such that dim P σ ′ = n − .Then ˜ F σ ′ = [ σ ˜ F σ , (20) where the union goes over all σ such that P σ ′ is a facet of P σ . Since the boundary ∂ ∆ n, consists of hypersimplices ∆ n − , and simplices ∆ n − , which correspond in G n, to the Grassmannians G n − , and complex projectivespaces C P n − , altogether way we obtain: Theorem 6. The universal space of parameters F n for T n -action on G n, is givenby the formal union F n = [ dim P σ = n − Pσ ∩ ◦ ∆ n, = ∅ ˜ F σ . (21)29 Universal space of parameters and admissible poly-topes Let F be a universal space of parameters for G n, and ˜ F σ the virtual space ofparameters for a stratum W σ . For x ∈ ◦ ∆ n, denote by ˜ F x = [ x ∈ ◦ P σ ˜ F σ . Theorem 7. ˜ F x = F n for any x ∈ ◦ ∆ n, . In order to prove this theorem, by Theorem 5, we need to prove that ˜ F σ ⊂ ˜ F x forany P σ such that dim P σ = n − and P σ ∩ ◦ ∆ n, = ∅ . G , It follows from Theorem 5 that the admissible polytopes of dimension are givenby the intersection with ∆ , of the planes x i + x j = 1 , where ≤ i < j ≤ .Without loss of generality because of the action of the symmetric group S , fix theadmissible polytope defined by x + x = 1 , denote it by P , the correspondingstratum by W and and its virtual space of parameters by ˜ F . The stratum W belongs to the chart M and it writes as b a b , a , b , b = 0 . It follows that ˜ F ∼ = (1 : 0) × C P × (1 : 0) .Let x ∈ ◦ ∆ , . The following cases are possible:1. If x ∈ ◦ P then ˜ F ⊂ ˜ F x .2. If x / ∈ ◦ P then x + x > or x + x < .30f x + x > then x belongs to the admissible polytope, we denote it by P +14 defined by the half-space x + x ≥ . The stratum which corresponds to thispolytope is, in the chart M , given by b a b b , a , b , b , b = 0 , so the corresponding virtual space of parameters is ˜ F +14 ∼ = (0 : 1) × C P × (1 : 0) .Since ˜ F +14 = ˜ F , it follows that ˜ F ⊂ ˜ F x .If x + x < then x belongs to the admissible polytope, we denote it by P − defined by the half-space x + x ≤ . The corresponding stratum is, in the chart M , given by a b a a b , a , a , a , b , b = 0 . The corresponding space of parameters is ˜ F − ∼ = (0 : 1) × C P A × (1 : 0) and wehave that ˜ F − ⊂ ˜ F x . We need to prove that the points A = (0 : 1) × (0 : 1) × (1 :0) , A = (0 : 1) × (1 : 0) × (1 : 0) and A = (0 : 1) × (1 : 1) × (1 : 0) belong to ˜ F x . In order to do that we consider the following cases.1) a) If x + x ≤ then x belongs to the intersection of half-spaces x + x ≤ and x + x ≤ . Consider the stratum given by b a a b , a , a , b , b = 0 . Its admissible polytope is exactly given by the intersection of these half-spacesand its virtual space of parameters coincides with its space of parameters which isthe point A . Thus, in this case A ∈ ˜ F x .31) If x + x ≥ then x + x + x ≤ and we consider the stratum given by a b a a , a , a , a , b = 0 . Its admissible polytope is given by x + x ≥ , its virtual space of parameters is (0 : 1) × (0 : 1) × C P and it contains the point A . It follows that A ∈ ˜ F x .2) a) If x + x ≤ the point x belongs to the intersection of half-spaces x + x ≤ and x + x ≤ . Now consider the stratum a b a b , a , a , b , b = 0 . It admissible polytope is exactly given by the intersection of these half-spaces andits virtual space of parameters coincides with its space of parameters that is withthe point A = (0 : 1) × (1 : 0) × (1 : 0) . Thus, A ∈ ˜ F x .b) if x + x ≥ it follows that x + x + x ≤ , so we consider the stratum a a a b , a , a , a , b = 0 . Its admissible polytope is exactly given by x + x ≥ and its virtual space ofparameters is C P × (1 : 0) × (1 : 0) . So, A ∈ ˜ F x .3) a) If x + x ≤ then x belongs to the intersection of the half-spaces x + x ≤ and x + x ≤ and we consider the stratum a b a a b , a , a , a , b , b = 0 , a b = a b . A = (0 :1) × (1 : 1) × (1 : 0) . Thus, A ∈ ˜ F x .b) If x + x ≥ the x + x + x ≤ . The stratum W +35 whose admissiblepolytope is defined in this way does not belong to the chart M . This stratumbelongs to the chart M and in this charts it writes as a 00 1 a a b , a , a , a , b = 0 . Its virtual space of parameters is C P × (1 : 0) × (1 : 0) . The transition functionbetween the charts M and M induces the homeomorphism f : F → F which can be extended to the homeomorphism ˜ f : F → F . This homeomor-phism is given by (( c : c ′ ) , ( c : c ′ ) , ( c : c ′ )) → (( c : c − c ′ ) , ( c : c − c ′ ) , ( c ( c − c ′ ) : c ( c − c ′ )) . It implies that the preimage of C P × (1 : 0) × (1 : 0) is given by (( c : c ′ ) , (1 :1) , ( c : c ′ )) , where ( c : c ′ ) ∈ C P . Thus, for c = 0 we obtain that thepoint A = (0 : 1) × (1 : 1) × (1 : 0) belongs to ˜ F x .Altogether we proved in this way that ˜ F ⊂ ˜ F x .By the action of the symmetric group we obtain this to be true for all ˜ F ij , that is ˜ F x = F . Using the same pattern the following holds as well: Proposition 12. If P σ , P σ ′ are admissible polytopes such that ◦ P σ ∩ ◦ P σ ′ has non-empty intersection with ◦ ∆ n, then ˜ F σ ∩ ˜ F σ ′ = ∅ .Proof. We differentiate the following cases depending on the dimensions of P σ and P σ ′ .1. Let dim P σ = dim P σ ′ = n − . We provide the proof for the case when each of P σ and P σ ′ is defined by just one half-space according to Theorem 2. In the case33hen P σ or P σ ′ is given as the intersection of the larger number of half spacesthe proof goes in an analogous way. Now, because of the action of the symmetricgroup we can assume that P σ is defined by the half-space x + x + . . . + x k ≤ , ≤ k ≤ n − and let P ′ σ is defined by x p + . . . + x p s ≤ . Since x + . . . + x n = 2 and these two half-spaces intersect it follows that there exists i , k + 1 ≤ i ≤ n such that i = p , . . . p s . Without loss of generality we can assume that i = k + 1 .It follows that the strata W σ and W σ ′ belong to the chart M ,k +1 . The stratum W σ writes in this charts as a a ... ... a k 00 1 a k +2 b k +2 ... ... a n b n , where a i = 0 , ≤ i ≤ n , i = k + 1 and b i = 0 , k + 2 ≤ i ≤ n . It follows that ˜ F σ = ( C P ) k − × (1 : 0) n − k − × ( C P ) k − × (1 : 0) n − k − × . . . × C P × (1 : 0) n − k − (22) × (1 : 0) n − k − × ( C P A ) ( n − k − n − k − . The stratum W σ ′ is, in this chart, given by the conditions b ′ p s = 0 , ≤ s ≤ l . Nowif ˜ F σ ∩ ˜ F ′ σ = ∅ than in ˜ F σ ′ we have ( c pq : c ′ pq ) = (1 : 0) or ( c pq : c ′ pq ) = C P for any ≤ p ≤ k and any k + 2 ≤ q ≤ n . In both cases we must have that b ′ p = 0 which implies that { , . . . , k } ⊂ { p , . . . , p l } . Moreover, there must exist p s / ∈ { , . . . , k } and note that not all b ′ i -s are zeros. We can assume that b ′ k +2 = 0 and b ′ k +3 = 0 . It follows that in ˜ F σ ′ we have that ( c k +2 ,k +3 : c ′ k +2 ,k +3 ) = (0 : 1) which together with (22) gives the contradiction with an assumption that ˜ F σ and ˜ F σ ′ intersect.2. Let dim P σ = n − and dim P σ ′ = n − . Assume that P σ is given by x + . . . + x k ≤ , ≤ k ≤ n − and P σ ′ is given by x p + . . . + x p l = 1 , ≤ l ≤ n − . Since these polytopes intersect it follows that we can assume that k + 1 / ∈ { p , . . . , p l } and that p / ∈ { , . . . k } . It follows that the both strata W σ W σ ′ belong to the chart M k +1 ,p . Then stratum W σ is given in this chart by a b a b ... ... a k b k a k +2 b k +2 ... ... a p +1 b p +1 ... ... a n b n , where a i b i = a j b i for ≤ i < j ≤ k and a i , b i = 0 for i ≥ k + 2 . It follows that ˜ F σ = (1 : 1) k − × ( C P A ) n − k − × (1 : 1) k − × ( C P A ) n − k − × . . . × (1 : 1) × ( C P A ) n − k − (23) × ( C P A ) n − k − × (( C P A ) n − k − ) ( n − k − n − k − . The points of the stratum W σ ′ in this chart satisfy conditions a ′ p s = 0 for ≤ s ≤ l . It follows that b ′ p s = 0 and b ′ q = 0 for q = p s , ≤ s ≤ l , see Proposition 6.Note that there exists q = k + 1 such that q / ∈ { p , . . . , p l } . It follows that ( c q p s : c ′ q p s ) = (0 : 1) or (1 : 0) in ˜ F ′ σ . Comparing to (23) we conclude that ˜ F σ ∩ ˜ F σ ′ = ∅ .3. Let dim P σ = dim P σ ′ = n − and assume that P σ is given by x + . . . + x k = 1 , ≤ k ≤ n − and P σ ′ is given by x p + . . . + x p l = 1 , ≤ l ≤ n − . We canassume that x / ∈ { p , . . . , p l } and p / ∈ { , . . . , k } . Then the strata W σ and W σ ′ M p The stratum W σ is in this chart given by a ... ... a k b k +1 ... ... b p +1 ... ... b n , wher b i = 0 , ≤ i ≤ k , b j = 0 , k + 1 ≤ i < j ≤ n , j = p . It follows that ˜ F σ = ( C P ) k − × (1 : 0) n − k − × . . . × C P × (1 : 0) n − k − (24) × (1 : 0) n − k − × ( C P ) ( n − k − n − k − . The points of the stratum W σ ′ are in this chart satisfy the conditions a ′ p s = 0 forall ≤ s ≤ l . It follows that b ′ p s = 0 and b ′ q = 0 for q = p s , ≤ s ≤ l . Let q be such that q / ∈ { , . . . , k } and q / ∈ { p , . . . , p l } . Note that there exist i , ≤ i ≤ k such that i ∈ { p , . . . p l } . It implies that ( c i q : c ′ i q ) = (0 : 1) in ˜ F ′ σ which together with (24) implies that ˜ F σ ∩ ˜ F σ ′ = ∅ . Corollary 8. Let x ∈ ◦ ∆ n, . If ˜ F σ , ˜ F σ ′ ⊂ ˜ F x then ˜ F σ ∩ ˜ F σ ′ = ∅ . ∆ n, Consider the hyperplane arrangement in R n given by A n : Π ∪ { x i = 0 , ≤ i ≤ n } ∪ { x i = 1 , ≤ i ≤ n } , (25)where the set Π is given by (9), and the face lattice L ( A n ) of the hyperplanearrangement A n . This lattice consists of the hyperplanes from A n and all inter-sections of the elements from A n . 36he hyperplane arrangement A n induces the hyperplane arrangement in R n − = { ( x , . . . , x n ) ∈ R n : x + . . . + x n = 2 } which is obtained by intersecting this R n − with the planes from (25).Denote by L ( A n, ) = L ( A n ) ∩ ∆ n, . Then L ( A n, ) provides decomposition for ∆ n, which we call chamber decomposition and for an element C ∈ L ( A n, ) wesay to be a chamber. Lemma 11. Let C ∈ L ( A n, ) such that dim C = n − . If C has a nonemptyintersection with ◦ P σ then C ⊂ ◦ P σ . Thus, C can be be obtained as the intersectionof the interiors of all admissible polytopes which contain it.Proof. We first note that if ◦ P σ ∩ C = ∅ then dim P σ = n − as well. Moreover,any facet of P σ which intersects ◦ ∆ n, belongs to some of the hyperplanes fromthe set Π which define the chamber decomposition C . For the second statement,we note that any wall V of the chamber C is contained in a facet of an admissiblepolytope which contains the chamber C . This follows from the fact that any of thehyperplanes from Π divides the hypersimplex ∆ n, into two admissible polytopes.The same is true for an arbitrary chamber from L ( A n, ) . Lemma 12. Any element C ∈ L ( A n, ) can be obtained as the intersection of theinteriors all admissible polytopes which contain C .Proof. We have that L ( A n, ) = L ( A ) ∩ ∆ n, . So, if C ∈ L ( A n, ) it follows that C = ∩ π i ,...i p for some planes π i ,...i p ∈ Π . Since π i ,...,i p ∩ ◦ ∆ n, is an admissiblepolytope and the planes from Π define the chamber decomposition L ( A n, ) thestatement follows.Therefore we have: Proposition 13. The chamber decomposition L ( A n, ) coincides with the decom-position of ◦ ∆ n, given by the intersections of interiors all admissible polytopeswhich are inside ∆ n, . Further on we denote the chambers from L ( A n, ) by C ω , where ω consist of thoseadmissible sets σ such that C ω ⊂ ◦ P σ . 37 .1 Moment map and the chamber decomposition Let C ω ∈ L ( A n, ) be a chamber, that is dim C ω = n − . According to Proposi-tion 13 we have that C ω = ∩ σ ∈ ω ◦ P σ and obviously dim P σ = n − . It followsfrom [7] that the spaces ˆ µ − ( x ) = ∪ σ ∈ ω F σ ⊂ G n, /T n are smooth manifolds andthey are diffeomorphic for all x ∈ C ω , that is diffeomorphic to some manifold F ω .On the other hand, as it is showed in [7] there exist the canonical homeomorphisms h σ : W σ /T σ → ◦ P σ × F σ given by h σ = (ˆ µ σ , p σ ) , where ˆ µ σ : W σ /T σ → ◦ P σ isinduced by the moment map ˆ µ : G n, /T n → ∆ n, , while p σ : W σ /T σ → F σ isinduced by the natural projection G n, → G n, / ( C ∗ ) n . For the main stratum W we have W/T n ∼ = ∆ n, × F , so it follows that F ⊂ F ω . Let ˆ C ω = ˆ µ − ( C ω ) . Corollary 9. For any C ω ∈ L ( A n, ) such that dim C ω = n − there existscanonical homeomorphism h C ω : ˆ C ω → C ω × F ω . where the manifold F Cω is a compactification of the space F n given by the spaces F σ such that C ω ⊂ P σ F ω = [ C ω ⊂ ◦ P σ F σ . (26)In some sense this can be generalized to an arbitrary chamber. As noted in [12],perceiving the spaces ˆ µ − ( x ) , ˆ µ − ( y ) as symplectic quotients it follows that theyare homeomorphic, that is to some space F ω , for any chamber C ω ∈ L ( A n, ) andany two points x, y ∈ C . When dim C ω = n − we deduce the following: Lemma 13. For any C ω ∈ L ( A n, ) , dim C ω = n − there exists canonicalhomeomorphism h C ω : ˆ C ω → C ω × F ω . The space F ω is a compactification of the space F and this compactification isgiven by the spaces F σ such that C ω ⊂ ◦ P σ and dim P σ = n − and a point, thatis the space F σ such that C ω ⊂ ◦ P σ and dim P σ = n − .Proof. It is obvious that if C ω ⊂ ◦ P σ and dim P σ = n − then F σ ⊂ F ω . Since C ω is of dimension n − there exists unique admissible polytope P σ , dim P σ = n − such that C ω ⊂ ◦ P σ , in fact P σ is defined by the underlying hyperplane for C ω .Thus, using Proposition 6 the statement follows.38ince the only interior admissible polytopes are of dimension n − or n − , , foran arbitrary C ω ∈ L ( A n, ) i of dimension ≤ n − we repeat the argument and inan analogous way deduce the following: Lemma 14. For any C ω ∈ L ( A n, ) , dim C ω ≤ n − there exists canonicalhomeomorphism h C ω : ˆ C ω → C ω × F ω . The space F ω is a compactification of F given by the spaces F σ such that C ω ⊂ ◦ P σ and dim P σ = n − and q points where q ≥ is the number of polytopes P σ suchthat C ω ⊂ ◦ P σ and dim P σ = n − . Note that the permutation action of the symmetric group S n on R n induces S n -action on A n , which further gives S n -action on the chambers of the chamber de-composition L ( A n, ) . Together with Corollary 1 we obtain: Corollary 10. The spaces F ω and s ( F ω ) are homeomorphic for any s ∈ S n andany C ω ∈ L ( A n, ) . Altogether we conclude: Proposition 14. A manifold F ω is a compactification of the space F ⊂ ( C P A ) N given by the equations (15) . This compactification is given by the spaces F σ suchthat C ω ⊂ ◦ P σ . Moreover, a space F σ is a point or it is obtained by restricting thehypersurfaces (15) to some factors ( C P B ) q ⊂ ( C P A ) N , where B = { (1 : 0) , (0 :1) } and ≤ q ≤ l , n − ≤ l ≤ N . Let F n be an universal space of parameters for G n, and consider the chart M .We assigned to any stratum W σ from G n, the virtual space of parameters ˜ F σ, asdescribed in [7] and in the previous sections. Moreover, for any ˜ F σ, it is definedthe projection p : ˜ F σ, → F σ , where F σ is the space of parameters for thestratum W σ .For C ω ∈ L ( A n, ) from Proposition 12 one directly deduces the following Corollary 11. Let C ω ∈ L ( A n, ) . Then ˜ F σ ∩ ˜ F ¯ σ = ∅ for any admissible sets σ, ¯ σ such that C ω ⊂ ◦ P σ . ◦ P ¯ σ . M ij for G n, in 3.2. Together with Theorem 4we obtain: Corollary 12. The union F n = [ C ω ⊂ ◦ P σ ˜ F σ . (27) is a disjoint union for any C ω ∈ L ( A n, ) . Therefore, for any chamber C ω and anychart M ij ⊂ G n, it is defined the projection p C ω ,ij : F n → F ω by p C ω ,ij ( y ) = p σ,ij ( y ) . where y ∈ ˜ F σ,ij . G n, /T n Let ◦ G n, /T n = ˆ µ − ( ◦ ∆ n, ) and let ˆ C ω = ˆ µ − ( C ω ) be as before. Theorem 8. The following disjoint decomposition holds: ◦ G n, /T n ∼ = [ ω ˆ C ω ∼ = [ ω ( C ω × F ω ) . (28) where the topology on the right hand side is given by the induced moment map ˆ µ : G n, /T n → ∆ n, and the natural projection G n, /T n → G n, / ( C ∗ ) n . The symmetric group S n acts on ◦ G n, /T n by s ( C ω × F ω ) = s ( C ω ) × s ( F ω ) ,that is C ω × F ω is homeomorphic to s ( C ω × F ω ) , which significantly simplifiesthe description of the elements in the union (28). This means that there exists l = l ( n ) ≥ and ω , . . . , ω l - the indices of the S n -action orbits representativessuch that ◦ G n, /T n = l [ i =1 [ s ∈ S n s ( C ω i × F ω i ) . (29) Example 9. For n = 4 , the chambers in ◦ ∆ , are of dimensions , , , and the S -action has one orbit in each of these dimensions, that is l (4) = 4 . Altogether, G n, /T n ∼ = ◦ G n, /T n ∪ ( n [ q =1 G n − , ( q ) /T n − ) ∪ ( n [ q =1 ∆ n − ( q )) . (30)40he topology on the right hand side of (30) is defined by the canonical embeddings C P n − ( q ) → G n, and G n − , ( q ) → G n, , ≤ q ≤ n .The universal space of parameters for a C P n − ( q ) , ≤ q ≤ n is a point. Thecanonical embeddings i q : G n − , ( q ) → G n, are defined by the inclusions C n − → C n , ( z , . . . , z n − ) → ( z , . . . z q − , , z q , . . . z n − ) , ≤ q ≤ n . Therefore, it isstraightforward to relate the universal spaces of parameters for G n, and G n − , ( q ) , ≤ q ≤ n . Proposition 15. The universal space of parameters F n − ,q for G n − , ( q ) ⊂ G n, , ≤ q ≤ n can be obtained from the universal space of parameters F n for G n, by the restriction: F n − ,q = F n | { ( c ij : c ′ ij ) ,i,j = q } , (31) which defines the projection r q : F n → F n − ,q . It follows that all previous constructions apply to F n − ,q and ∆ n − , ( q ) ⊂ ∂ ∆ n, obtained as ∆ n − , ( q ) = ∆ n, ∩ { x q = 0 } , ≤ q ≤ n . Denote by p qC ω ,ij : F n − ,q → F ω the map given by Corollary 12 for the Grassmannian G n − , ( q ) , ≤ q ≤ n . In this way we obtain: Corollary 13. For any chamber C ω ⊂ ∂ ∆ n, and any chart M ij it is defined theprojection p C ω ,ij : F n → F ω . If C ω ⊂ ∆ n − ( q ) this projection maps F n to a point,while for C ω ⊂ ∆ n − , ( q ) it is defined by p C ω ,ij ( y ) = ( p qC ω ,ij ◦ r q )( y ) . So, let us now consider the space U n = ∆ n, × F n . (32)It can be inductively defined the projection from U n using the following pattern: U n → ( ◦ ∆ n, ×F n ) ∪ ( n [ q =1 U n − ,q ) ∪ ( n [ q =1 ∆ n − ( q )) , (33)for U n − ,q = ∆ n − , ( q ) × F n − ,q , which is an identity for x ∈ ◦ ∆ n, , it is given by ( x, f ) → ( x, r q ( f )) if x ∈ ∆ n − , ( q ) , while ( x, f ) → x if x ∈ ∆ n − ( q ) , where ≤ q ≤ n .Therefore, from Corollary 12, Theorem 8 and Corollary 13 we obtain:41 heorem 9. For any chart M ij ⊂ G n, the map G : U n → G n, /T n , G ( x, y ) = h − C ω ( x, p C ω ,ij ( y )) if and only if x ∈ C ω (34) is correctly defined. Then from Theorem 8, formula (30) and Proposition 15 we deduce: Theorem 10. The map G is a continuous surjection and the orbit space G n, /T n is homeomorphic to the quotient of the space U n by the map G . The more explicit proof proceeds in an analogous way as the proof of Theorem11.1 in [7], which describes the orbit space G , /T . References [1] A. 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