Singularities on Flag Spaces ^{∗}

Miloš Božek (Bratislava) and Konrad Drechsler (Halle)

September 4, 1998

Abstract We generalized in 1980 the classical concept of flag spaces by replacing the determining linear order by a partial one. Such new flag spaces are also algebraic varieties in a natural way, but they can contain singularities in contrast to the classical ones. Our aim is to characterize when it happens only by properties of the partial order. We are sure that the following conjecture holds: A flag space has singular points if and only if its partially ordered set contains a crown. But we are not able to prove it without using some additional condition.

The classical concept of flag spaces was generalized in severel manners using a partially ordered set. One possibility is to join the elements of a system of flags according to the relations of the order, see e.g. [G], [L]. Our idea was to replace the determining linear order by a partial one [BD1]. Such new flag spaces are also algebraic varieties in a natural way, but they can contain singularities in contrast to the classical ones. Our aim is to characterize when it happens only by properties of the partial order.

An interesting property of posets is their dismantlability [DR]. We have already remarked in [BD1], Th. 3, that flag spaces corresponding to dismantlable posets are smooth. We shall prove it in detail in Section 4 of the present paper. For this purpose we extend our considerations from [BD1] by constructing regular algebraic local parametrizations of those flag spaces adapted to the fibrations which are induced by doubly irreducible elements of the corresponding poset.

Duffus and Rival proved [DR] that a partially ordered set is dismantlable if and only if it contains no crown. This connection enables us to find singularities on flag spaces in Section 5. Moreover, we are sure that the following conjecture holds: A flag space has singular points if and only if its partially ordered set contains a crown. But we are not able to prove it without using some additional condition (Theorem 5.1). Nevertheless, we computed many examples in which that condition is not fulfilled but the flag space is singular, and we have no counterexample. So it could be a good conjecture.

∗It will be published in Mathematica Slovaka

1 Generalized flag spaces

Let (M, ⊏) be a finite partially ordered set, let (M̂, ⊏) be its closure by an additional lower and upper bound 0 and 1 and let

δ : (M̂, ⊏) → (ℕ, <) (1)

be a strictly order-preserving mapping^{1} into the naturally ordered set ℕ of non-negative integers such that δ0= 0 and n = δ1. Then we call τ = (M̂, ⊏, δ) a type. Let K^{n} be the n-dimensional vector space over a field K and let (L, ⊂) be the lattice of its subspaces.

Definition 1.1: The set of all order-preserving mappings γ from τ into (L, ⊂) such that δm is equal to the dimension of the subspace γm we call a flag space F (τ) = F (K, τ) over K of the type τ = (M̂, ⊏, δ). Its elements are called flags in K^{n}.

An algebraic set F_{τ} corresponds to a flag space F (τ) in the product

G(M, δ) = ∏_{m∈M} G^{m}(δm, n) (2)

of Grassmann manifolds G^{m}(δm, n) of δm-subspaces of the vector space K^{n} in a natural way: the equations defining F_{τ} are given by the pairs coming from the relation ⊏. We shall not make any distinction between F_{τ} and F (τ). Let a type ω = (M̂, ≺, δ) be a refinement of τ = (M̂, ⊏, δ) which means that ⊏ is a subset of ≺. F (ω) can be considered as an algebraic subset of F (τ).

1In [BD1] we demanded only (M̂, ⊑) → (ℕ, ≤) to be order-preserving. In [BD3] we have called the type defined in the present paper a reduced type and we have remarked there that to every flag space there exists af flag space with reduced type that is naturally equivalent to the given one. All assertions in the present paper hold for a reduced type only.

Definition 1.2: We call ω a linearization of τ on a subset C ⊆ M if the restriction ≺_{C} of ≺ to C × C is a linear (total) order on C. A flag γ ∈ F (τ) is said to be linear on C if γ ∈ F (ω) where ω is a linearization of τ on C, and it is said to be linear if it is linear on M.

2 Order theoretic preliminaries

Let −⊏ denote the set of all pairs of neighbours of the partially ordered set (M, ⊏). Since M is finite, ⊏ is generated by the pairs of −⊏ as the transitive closure. Let ℤ/s be the (additive factor) group of integers modulo s.

Definition 2.1: Let s ∈ ℕ and s ≥ 4. A map ζ : ℤ/s → (M, ⊏) is said to be a cycle in (M, ⊏) if either ζi −⊏ ζ(i+1) or ζ(i+1) −⊏ ζi for all i ∈ ℤ/s.

Let max ζ or min ζ denote the set of all maxima or minima of ζ respectively and let exζ = max ζ ∪ min ζ. Finally, let r_{ζ} = #(max ζ) = #(min ζ) the cardinality. A cycle ζ determines a partial order ⊏^{ζ} on its image imζ transitively generated by the pairs of neighboured values of ζ. ⊏^{ζ} does not need to coincide with the restriction ⊏_{imζ}.

Definition 2.2: A cycle ζ in (M, ⊏) is said to be a crown cycle contained in (M, ⊏) if it fulfils the following conditions:

(i) r_{ζ} ≥ 2,

(ii) ζ is injective and

(iii) ⊏^{ζ}=⊏_{imζ}.

The partially ordered set (exζ, ⊏_{exζ}) is then called a crown of order 2r_{ζ} contained in (M, ⊏).

Examples 2.3: We shall describe the partially ordered set (M, ⊏) by the Hasse diagram. By ∘ we denote the elements of exζ and by • the remaining ones.

[figure: 0; 1; 2; 3; 4; 5; 6; 7; 0; 1; 2; 3; 0; 1; 2; 3; 4; 5; 6; 7; 0; 1; 2; 3; 4; 5]
s = 8  s = 4  s = 8  s = 6
r_{ζ} = 2  r_{ζ} = 1  r_{ζ} = 2  r_{ζ} = 2
Fig. 1  Fig. 2  Fig. 3  Fig. 4

The figure 1 illustrates a crown, the other figures do not describe crowns contained in (M, ⊏).

An element m in M is said to be doubly irreducible in (M, ⊏) if m has at most one lower and at most one upper neighbour in (M, ⊏).^{2} Let D(M, ⊏) denote the set of all doubly irreducible elements in (M, ⊏).

Definition 2.4: [DR] (M, ⊏) is called dismantlable (by doubly irreducible elements) if the elements of M can be labelled so that m^{ν} ∈ D(M \ {m^{1}, ⋯, m^{ν−1}}, ⊏) for every ν = 1, ⋯, #M.

Theorem 2.5: [DR] A finite partially ordered set is dismantlable if and only if it contains no crowns. For the proof see [DR] and [KR]. Note that the condition for a crown to be contained in (M, ⊏) is not strong there as our Definiton 2.2. But we need it for the purpose of the proof of our Theorem 5.1.

3 Coordinate description of flag spaces

In the rest of the paper the field K is assumed to be infinite.

Recall that any basis e_{1}, ⋯, e_{n} of the vector space K^{n} induces local coordinates g ←↦ (g_{νμ} : ν = 1, ⋯, k; μ = k + 1, ⋯, n) in the Grassmann manifold G(k, n). They are defined on the Zariski-open set G̃(k, n) ⊂ G(k, n) consisting of all g ∈ G(k, n) which are transversal to the subspace of K^{n} spanned by the last n − k vectors of the basis. The coordinates g_{νμ} are determined by the condition

2This notion comes from lattice theory: m is irreducible with respect to both lattice operations.

e_{ν} + ∑_{μ=k+1}^{n} g_{νμ}e_{μ} ∈ g. (3)

Using this coordinates we can identify G̃(k, n) with K^{k(n−k)}.

The affine Grassmann coordinates mentioned above induce in a natural way local coordinates (m_{νμ} : m ∈ M; ν = 1, ⋯, δm; μ = δm + 1, ⋯, n) in G(M, δ) defined on the Zariski-open set

G̃(M, δ) = ∏_{m∈M} G̃^{m}(δm, n)

for a flag space F (τ). Thus, varying the basis in K^{n} we get a covering of the flag space by Zariski-open subsets F̃(τ) = F (τ) ∩ G̃(M, δ) which are imbedded in the corresponding affine space G̃(M, δ) = K^{d}, where d = ∑_{m∈M} δm(n − δm).

To each relation m′ ⊏ m of (M, ⊏) there belongs the inclusion γm′ ⊂ γm of subspaces of K^{n} for all γ ∈ F (τ). Thus, using the relations (3), we get a set of polynomials

f_{m′m}^{νμ}(γ) = m_{νμ} − m′_{νμ} + ∑_{λ=δm′+1}^{δm} m′_{νλ}m_{λμ} , (4)

ν = 1, ⋯, δm′; μ = δm + 1, ⋯, n,

in the ideal I(τ) corresponding to F (τ) of the ring

R = K[m_{νμ} : ν = 1, ⋯, δm; μ = δm + 1, ..., n; m ∈ M]

of coordinates on G(M, δ). I(τ) is generated by the polynomials f_{m′m}^{νμ}(γ) for m′ −⊏ m only.

Let m′′ ⊏ m′ ⊏ m. It is easy to check that

f_{m′′m}^{νμ}(γ) = f_{m′′m′}^{νμ}(γ) + f_{m′m}^{νμ}(γ) + ∑_{λ=δm′′+1}^{δm′} m′′_{νλ}f_{m′m}^{λμ}(γ) − ∑_{λ=δm′+1}^{δm} f_{m′′m′}^{νλ}(γ)m_{λμ} (5)

for all ν = 1, ⋯, δm′′; μ = δm + 1, ⋯, n and for all γ ∈ F (τ).

We also use the notation

f_{mm′}^{νμ}(γ) = −f_{m′m}^{νμ}(γ). (6)

4 Smooth flag spaces

Let m̄ ∈ M and let π : F (M̂, ⊏, δ) → F (M̂ \ {m̄}, ⊏, δ) be the canonical projection.

Lemma 4.1: If m̄ is doubly irreducible in (M, ⊏) and if u and o are the neighbours of m̄ in (M̂, ⊏), then to every flag γ^{0} ∈ F (M̂, ⊏, δ) there exists a Zariski-neighbourhood F̃ and a Zariski-open set G̃ ⊆ G(δm̄ − δu, δo − δu) so that πF̃ is Zariski-open and F̃ is algebraically isomorphic to πF̃ × G̃.

Corollary 4.2: If m̄ is doubly irreducible in (M, ⊏) and if F (M̂ \ {m̄}, ⊏, δ) is smooth, then F (M̂, ⊏, δ) is smooth, too.

This yields the following

Theorem 4.3: Let K be infinite. If (M, ⊏) is dismantlable then the flag space F (M̂, ⊏, δ) is smooth for all δ.

Remark 4.4: With some more effort it can be shown that under the assumptions of Lemma 4.1 π is an algebraic locally trivial fibration with the fibre G(δm − δu, δo − δu). This is a strengthening of Theorem 1 in [BD1] from which we could obtain the above Corollary 4.2 and the Theorem 4.3 as a consequence.

Proof of Lemma 4.1: Let γ^{0} ∈ F (M̂, ⊏, δ) be arbitrary. Choose a basis in K^{n} such that γ^{0} ∈ F̃(M̂, ⊏, δ) and put F̃ = F̃(M̂, ⊏, δ) and F̃′ = F̃(M̂ \ {m̄}, ⊏, δ). Finally, choose a basis in K^{δo−δu} and put G̃ = G̃(δm̄−δu, δo−δu). The sets F̃, F̃′ and G̃ are Zariski-open, and πF̃ = F̃′ because of the assumption of Lemma 4.1.

Let φ : F̃ → G̃ be the map defined by coordinates

(φγ)_{νμ} = (γm̄)_{δu+ν δu+μ} (7)

ν = 1, ⋯, δm̄ − δu; μ = δm̄ − δu + 1, ⋯, δo − δu.

Then

Φ : F̃ → F̃′ × G̃, γ ↦ (πγ, φγ) (8)

is an algebraic map.

We shall construct Ψ = Φ^{−1} and shall show that it is an algebraic map. To every pair (γ′, g′) ∈ F̃′ × G̃ with coordinates

γ′ ←↦ (m_{νμ} : m ∈ M \ {m̄}; ν = 1, ⋯, δm; μ = δm + 1, ⋯, n),
g′ ←↦ (g′_{νμ} : ν = 1, ⋯, δm̄ − δu; μ = δm̄ − δu + 1, ⋯, δo − δu)

consider the linear subspace ψ(γ′, g′) = g ∈ G̃(δm̄, n) with the Grassmann coordinates

g_{νμ} = g′_{ν−δu μ−δu} if ν = δu + 1, ⋯, δm̄; μ = δm̄ + 1, ⋯, δo, (9)

g_{νμ} = u_{νμ} − ∑_{λ=δu+1}^{δm̄} u_{νλ}g′_{λ−δu μ−δu} (10)

if ν = 1, ⋯, δu; μ = δm̄ + 1, ⋯, δo,

g_{νμ} = o_{νμ} + ∑_{λ=δm̄+1}^{δo} g′_{ν−δu λ−δu}o_{λμ} (11)

if ν = δu + 1, ⋯, δm̄; μ = δo + 1, ⋯, n,

g_{νμ} = o_{νμ} + ∑_{λ=δm̄+1}^{δo}(u_{νλ} − ∑_{κ=δu+1}^{δm̄} u_{νκ}g′_{κ−δu λ−δu})o_{λμ} (12)

if ν = 1, ⋯, δu; μ = δo + 1, ⋯, n.

So we obtain an algebraic map Ψ : F̃′ × G̃ → G̃(M, δ) defined by

(13) (Ψ(γ′, g′))m = \{ \begin{matrix}ψ(γ′, g′) & if m = m̄, \\ γ′m & otherwise. \end{matrix}

To prove that Ψ(F̃′ × G̃) ⊆ F̃ it suffices to check that γ′u ⊂ ψ(γ′, g′) ⊂ γ′o because u and o are the only neighbours of m̄ in M̂. But this is fulfilled in virtue of (9) - (12). The equalities (9) together with (7),(8) and (13) imply that Ψ = Φ^{−1}. □

5 Singularities on flag spaces

Theorem 5.1: Let K be infinite and let a type τ = (M̂, ⊏, δ) fulfil the following conditions:

(a) (M, ⊏) contains a crown cycle ζ^{0}.

(b) There exists y −⊏^{ζ^{0}} x^{0} with δy = min δζ^{0} so that there does not exist a sequence of injective cycles (ζ^{j})_{j=1,··· ,k} with min ζ^{j} = {y} ^{3} and a sequence (x^{j})_{j=0,··· ,k} of distinct upper neighbours of y with y −⊏^{ζ^{j}} x^{j−1}, y −⊏^{ζ^{j}} x^{j} for all j = 1, ⋯, k and with y −⊏^{ζ^{0}} x^{k}.

Then every linear flag γ^{0} is a singular point in the corresponding flag space F (τ) for all dimension functions δ.

To prove the theorem we shall need several lemmas. We start by choosing of a basis in K^{n} such that γ^{0} is described by the zero point of the induced local coordinate system on G(M, δ). For every flag γ ∈ F (τ) we consider the K-vector space D(τ, γ) spanned by the differential forms

(14) df_{m′m}^{νμ}(γ) = dm_{νμ} − dm′_{νμ} + ∑_{λ=δm′+1}^{δm} dm′_{νλ}m_{λμ} + ∑_{λ=δm′+1}^{δm} m′_{νλ}dm_{λμ},
ν = 1, ⋯, δm′ ; μ = δm + 1, ⋯, n ; m′ −⊏ m.

From (5) it follows that for each cycle ζ with r_{ζ} = 1 and for all ν = 1, ⋯, min δζ; μ = max δζ + 1, ⋯, n each of the f_{ζi ζ(i+1)}^{νμ}(γ), i ∈ ℤ can be expressed in the ring R by the remaining ones belonging to ζ. Therefore

Lemma 5.2: If r_{ζ} = 1 then each of the df_{ζi ζ(i+1)}^{νμ}(γ), ν ≤ δm; μ > δm depends linearly of the remaining ones for all γ ∈ F (τ).

This is true also for any cycle and for the flag γ = γ^{0} by the following

Lemma 5.3: For every cycle ζ : ℤ/s → (M, ⊏) and for every pair (ν, μ) satisfying ν ≤ δm and μ > δm for all m ∈ imζ it holds that

∑_{i∈ℤ/s}df_{ζi ζ(i+1)}^{νμ}(γ^{0}) = 0.

Proof: Because ν ≤ δm, μ > δm and m ∈ imζ the pair (ν, μ) occurs in (4) for every m′ −⊏^{ζ} m. The equalities (6) and (14) yield

df_{ζi ζ(i+1)}^{νμ}(γ^{0}) = d(ζ(i + 1))_{νμ} − d(ζi)_{νμ}

3That means r_{ζ^{j}} = 1 and ζ^{j} is not a crown cycle.

for all i ∈ ℤ/s. □

According to the Jacobi criterion γ^{0} is a singular point of F (τ) if there exists a flag γ^{1}(t) ∈ F (τ), t trancendent over K, with γ^{1}(t) → γ^{0} for t → 0 such that

dim_{K} D(τ, γ^{0}) < dim_{K(t)} D(τ, γ^{1}(t)). (15)

First, we shall construct such a suitable flag γ^{1}(t). Let U_{x} be the set of all u ∈ M so that x ⊑ u and let U^{0} be the connected component of ⋃_{y−⊏x}U_{x}

which contains x^{0}. Let P^{0} be the set of all p ∈ M \ U^{0} for which there exists a u ∈ U^{0} so that p ⊏ u and δp > δy.

Definition 5.4: Let γ^{1}(t) be the point of G(M, δ) given by its coordinates:

(i) u_{δy+1 n}  = t^{n−δy−1} for all u ∈ U^{0},
(ii) u_{δp+1 n}  = −t^{n−δp−1} for all u ∈ U^{0} and p ∈ P^{0} with δp + 1 ≤ δu,
(iii) p_{δy+1 δp+1} = t^{δp−δy} for all p ∈ P^{0},
(iv) p_{δq+1 δp+1} = −t^{δp−δq} for all q, p ∈ P^{0}, δq < δp,
(v) m_{νμ}  = 0  otherwise.

Lemma 5.5: The point γ^{1}(t) is a flag of F (τ).

Proof: We have to check that all f_{m′m}^{νμ}(γ^{1}(t)) vanish identically in t.

Case 1: m′ −⊏ m = u′ −⊏ u; u′ ∈ U^{0}, u ∈ U^{0} : The equality (4) is interesting for μ = n only. The sum in (4) vanishes because λ ≠ n. So, it remains u_{νn} − u′_{νn}, ν = 1, ⋯, δu′, which equals zero because of (i), (ii) and (v).

Case 2: m′ −⊏ m = m′ −⊏ u; u ∈ U^{0}, m′ ∈ M \ U^{0} : If μ ≠ n then all u_{νμ} vanish and so does the sum, and also m′_{νμ} because δm′ + 1 < δu + 1 ≤ μ. If δm′ ≤ δy then m′_{νμ} equals 0 for all μ = m′ + 1, ⋯, n, and so does the sum. Also u_{νμ} = 0 because ν ≤ δm′ < δy + 1 < δp + 1 for all p ∈ P^{0}. If μ = n and δy < δm′ < δu, which means that m′ ∈ P^{0}, then it remains from (4) u_{νn} −m′_{νn} −m′_{ν δm′+1}t^{n−δm′−1} only according to (ii) and (v). It holds m′_{νn} = 0 for all ν because δm′ + 1 < n. If ν = δy + 1 then u_{νn} = t^{n−δy−1} and m′_{ν δm′+1} = t^{δm′−δy} in virtue of (i) and (iii). If ν = δq + 1, where q ∈ P^{0}, δq < δm′, then u_{νn} = −t^{n−δq−1} and m′_{ν δm′+1} = −t^{δm′−δq} because of (ii) and (iv).

Case 3: m′ −⊏ m = m′ −⊏ p, p ∈ P^{0}:

The equality (4) is interesting only for μ = δp + 1, and then it equals p_{ν δp+1} − m′_{ν δm′+1}t^{δp−δm′}. If there is an r ∈ P^{0} such that ν = δr + 1 then p_{ν δp+1} = −t^{δp−δr} and m′_{ν δm′+1} = −t^{δm′−δr} because of (iv). If ν = δy + 1 then p_{ν δp+1} = t^{δp−δy} and m′_{ν δm′+1} = t^{δm′−δy} because of (iii). Otherwise p_{ν δp+1} = m′_{ν δm′+1} = 0 according to (v).

Case 4: m′ −⊏ m = p −⊏ m, p ∈ P^{0}, m ∈ M \ (P^{0} ∪ U^{0}) : The condition (v) implies m_{ν μ} = 0 for all ν = 1, ⋯, δm, μ = δm + 1, ⋯, n. But then it remains only −p_{νμ} from (4). It holds μ > δm ≥ δp + 1, thus p_{νμ} = 0 again according to (v).

In all other cases the equality (4) is trivial because of the condition (v). □

Lemma 5.6: Let x ∈ U^{0} with y −⊏ x but x ≠ x^{0}. Then independently of the choice of γ, there exists a set of generators of D(τ, γ) which does not contain df_{yx}^{δy n}(γ).

Proof: Because of the definition of U^{0} there exists a sequence of injective cycles (ζ^{j})_{j=1,··· ,k} with min ζ^{j} = {y} and there exists a sequence (x^{j})_{j=0,··· ,k} of distinct upper neighbours of y with x^{k} = x and with y −⊏^{ζ^{j}} x^{j−1}, y −⊏^{ζ^{j}} x^{j} for all j = 1, ⋯, k. By Lemma 5.2 we can successively remove the differential forms df_{yx^{j}}^{δy n}(γ) if j ≠ 0. □

Assumption 5.7: In the following considerations we assume that the set of generators of D(τ, γ) does not contain df_{yx}^{δy n}(γ) for x ∈ U^{0} unless x ∈ imζ^{0}.

Now we consider the differential form

(16) df_{yx^{0}}^{δy n}(γ) = dx_{δy n} − dy_{δy n} + ∑_{λ=δy+1}^{δx^{0}} dy_{δy λ}x_{λn}^{0} + ∑_{λ=δy+1}^{δx^{0}} y_{δy λ}dx_{λn}^{0},

Because δy = min δζ^{0}, the conditions of Lemma 5.3 are fulfilled. Therefore, by Lemma 5.3 and Assumption 5.7 the differential form (16) for γ = γ^{0} is linearly dependent on the remaining generators of D(τ, γ^{0}).

Lemma 5.8: Let the type of the flag space satisfy the conditions of Theorem 5.1. Then the differential form (16) for γ = γ^{1}(t) is linearly independent over K(t) on the remaining generators of D(τ, γ^{1}(t))

Proof: The differential dy_{δy δy+1} occurs in df_{yx^{0}}^{δy n}(γ^{1}(t)) in the first sum only and there with the coefficient x_{δy+1 n}^{0} = t^{n−δy−1} because of Definition 5.4, (i).

We shall show that dy_{δy δy+1} occurs in the remaining generators of D(τ, γ^{1}(t)) only with a coefficient which vanishes identically in t. The differential dy_{δy δy+1} generally occurs in (14) if m = y or if m′ = y.

In the first case it occurs in the second sum because δm′ < δy. Its coefficient there is m′_{ν δy}. But m′ is not in U^{0} ∪ P^{0} for the same reason. Thus, m′_{ν δy} = 0 because of Definition 5.4, (v).

In the second case it occurs in the first sum because δy + 1 ≤ δm. Its coefficient is m_{δy+1 μ}. It is zero unless m ∈ U^{0} ∪ P^{0}. But m cannot sit in P^{0} because y −⊏ m ⊏ u for some u ∈ U^{0}. Therefore, m = x ∈ U^{0}. By Definition 5.4 we have x_{δy+1 μ} ≠ 0 only if μ = n . Then the considered differential form (14) is df_{yx}^{δy n}(γ). By Lemma 5.6 and by Assumption 5.7 it should not occur in the set of the generators unless x ∈ imζ^{0}. Then either df_{yx}^{δy n}(γ) = df_{yx^{0}}^{δy n}(γ) or the condition (b) of Theorem 5.1 is not fulfilled by virtue of the definition of U^{0}. □

Let D^{∗}(τ, γ) be the subspace of D(τ, γ) generated by all differential forms (14) other than (16). Because γ^{1}(0) = γ^{0} it holds that

dim_{K(t)} D^{∗}(τ, γ^{1}(t)) ≥ dim_{K} D^{∗}(τ, γ^{0}). (17)

Lemma 5.3 implies that

dim_{K} D(τ, γ^{0}) = dim_{K} D^{∗}(τ, γ^{0}) (18)

and from Lemma 5.8 it follows that

dim_{K(t)} D(τ, γ^{1}(t)) = dim_{K(t)} D^{∗}(τ, γ^{1}(t)) + 1. (19)

Therefore, (15) is true and Theorem 5.1 is proved. □

Remark 5.9: It could seem that we did not need the condition (a) in the proof. But notice this fact: If there be no crown cycle but a cycle in (M, ⊏), then condition (b) cannot be fulfilled. See figures 2, 3 and 4.

Remark 5.10: We conjecture that in Theorem 5.1 it suffices to suppose the flag γ^{0} to be linear on imζ^{0} only.

Example 5.11: Let n ≥ 4. The type τ = (M, ⊏, δ) given by Figure 5 fulfils condition (a) but does not fulfil condition (b) of Theorem 5.1.

[figure] [figure] [figure]

u

x^{0} x^{1}

y^{1} y^{0}

(M, ⊏) δ

3

2

1 Fig. 5 F (τ) consists of two irreducible algebraic components F (τ_{1}) and F (τ_{2}) both of which are flag spaces too. The types τ_{i} can be obtained from τ by identifying x^{0} = x^{1} and y^{0} = y^{1} resp.. The intersection F (τ_{1}) ∩ F (τ_{2}) is the space of all linear flags. Therefore, every linear flag is singular on F (τ).

The foregoing geometric considerations can be replaced by algebraic ones in terms of the polynomials (4).

There are many other examples in which (a) is fulfilled but (b) is not. In each of them, proved in a special way, the linear flags are singular. Nevertheless, a general proof remains still to be done.

References

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[BD2] M. Božek and K. Drechsler: Faserungen von Fahnenmengen. Beitr.Algebra Geom. 20 (1985), 203-213.

[BD3] M. Božek and K. Drechsler: Morphismen von Fahnenmengen. Acta Mathematica Universitatis Comenianae 58/59 (1990), 135-148.

[DR] D. Duffus and I. Rival: Crowns in dismantlable partially ordered sets. Combinatorics. (Proc. Colloq. Keszthely, 1976), colloq. Math. Janos Bolyay 18 (1976), 271-292.

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