Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 479-488.

The Second Extension of the Thas-Walker Construction

Rolf Riesinger

Patrizigasse 7/14, A–1210 Vienna, Austria

Abstract. We extend the extended Thas-Walker construction by introducing the concept “flocklet” of the Klein quadric. Thus we have a tool to construct spreads with asymplecticly complemented regulization. MSC 2000: 51A40 Keywords: spread, flock, flocklet, flockling

1. Introduction

Independently, M. Walker [15] and J. A. Thas (unpublished) discovered that to each flock of an elliptic or hyperbolic quadric or of a quadratic cone of PG(3, q) a spread of PG(3, q) and, consequently, a (finite) translation plane is constructable; compare [1, p. 8], [12, p. 441] or the surveys [3], [13, p.95-96], [14]. The Thas-Walker construction remains valid for flocks of PG(3, 𝕂) with arbitrary commutative field 𝕂; cf. [6, p. 146–149]. In [7] the author used flocks of PG(3, ℝ) to construct spreads representing topological translation planes. The extended Thas-Walker construction exhibited in [8], hereafter called [ETW], starts with a flockoid of a Lie quadric of PG(4, 𝕂) and yields also a spread; a Lie quadric L_{4} is a hyperquadric of a Pappian projective 4-space such that L_{4} has no vertex and contains a line; a collection D of conics contained in a Lie quadric L_{4} of a Pappian projective 4-space is called a flockoid of L_{4}, if the following two conditions hold: (FD1) For each generatrix g of L_{4} there exists exactly one conic k ∈ D with g ∩ k ≠ ∅. (FD2) There are at most two improper conics in D. In view of the present article, the paper [ETW] deals with the first extension of the Thas-Walker construction.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

Let Π = (P, L) be a Pappian projective 3-space with point set P and line set L. As we deal with spreads composed of reguli and at most two exceptional lines, so we standardize by defining: A proper regulus R is the set of lines meeting three mutually skew lines; the directrices of R form the complementary (opposite) regulus R^{c}; if x ∈ L, then {x} is called an improper regulus; {x}^{c} := {x}.

Definition 1. Let S be a spread of Π and let Σ be a collection of (proper or improper) reguli contained in S. We call Σ a regulization of S, if the following hold: (RZ1) Each line of S belongs either to exactly one regulus of Σ or to all reguli of Σ. (RZ2) There are at most two improper reguli in Σ. The set ∪(R^{c}|R ∈ Σ) =: S_{Σ}^{c} is named complementary congruence of S with respect to Σ. If S_{Σ}^{c} is an elliptic linear congruence of lines, then Σ is called an elliptic regulization of S. If S_{Σ}^{c} belongs to a linear complex of lines, then we say that Σ is a symplecticly complemented regulization, otherwise we speak of an asymplecticly complemented regulization. If S_{Σ}^{c} belongs to a single linear complex of lines, then Σ is called a unisymplecticly complemented regulization.

The papers [ETW], [9], and [10] are devoted to the construction and investigation of spreads with symplecticly complemented regulization. For the real projective 3-space PG(3, ℝ) an example of a non-regular spread with an asymplecticly complemented regulization is given in [6, (4.1.7)].

Let λ be the well-known Klein mapping of L onto the Klein quadric H_{5} which is embedded into a projective 5-space Π_{5} with point set P_{5}; cf. e.g. [4] and the translation table in [2, p. 29–30]. A Latin <Greek> plane on H_{5} is the λ-image of a star of lines <a ruled plane>. If R is a proper or improper regulus, then λ(R) is an irreducible conic or a point. For obvious reasons, we speak of proper or improper conics. If S is a spread of Π with the unisymplecticly complemented regulization Ω, then {λ(R^{c})|R ∈ Ω} is a flockoid of a uniquely determined Lie quadric L_{4} with L_{4} ⊂ H_{5}; cf. [ETW, Prop. 1].

Recall the extended Thas-Walker construction [ETW, Prop. 2]: If D is a flockoid of a Lie quadric L_{4} with L_{4} ⊂ H_{5}, then ∪((λ^{−1}(k))^{c}| k ∈ D) is a spread of Π with the regulization {(λ^{−1}(k))^{c}|k ∈ D} which is either unisymplecticly complemented or elliptic.

In Proposition 1 we start with a spread S of Π admitting an asymplecticly complemented regulization^{1} Γ and investigate the set {λ(R^{c})|R ∈ Γ} =: C of conics. In the proof of Proposition 1 we shall find that C is a “flocklet” of the Klein quadric H_{5}; we define the concept “flocklet”, as follows.

Definition 2. A collection B of (proper or improper) conics contained in the Klein quadric H_{5} is called a flocklet <flockling> of H_{5}, if the following two conditions hold: (FT1)<(FG1)> For each Latin <Greek> plane γ on H_{5} there exists exactly one conic k ∈ B with γ ∩ k ≠ ∅. (FT2)<=(FG2)> There are at most two improper conics in B.

Depending on the intersection M of the carrier planes of the participating proper conics we distinguish three types of flocklets:

1The assumption in Proposition 1 is more comprehensive; cf. Remark 1 and Definition 3.

1) M is empty (unbundled flocklet)

2) M consists of exactly one point (star flocklet)

3) M is a line (linear flocklet)^{2}. In the same way unbundled flocklings, star flocklings, and linear flocklings are defined.

The second extension of the Thas-Walker construction starts with a flocklet A of the Klein quadric H_{5}. Then ∪((λ^{−1}(k))^{c}|k ∈ A) is a spread of Π admitting the regulization {(λ^{−1}(k))^{c}|k ∈ A} which is either asymplecticly complemented or unisymplecticly complemented or elliptic; cf. Proposition 2 and Remark 1. Each flockoid of a Lie quadric L_{4} can be interpreted as flocklet and also as flockling of a Klein quadric H_{5} containing L_{4}; cf. Remark 3. Each flock of an elliptic quadric Q_{e} can be interpreted as flocklet and also as flockling of a Klein quadric H_{5} containing Q_{e}; cf. Remark 4. Note, that a flock of a quadric Q covers Q, but a flocklet of a Klein quadric H_{5} is no covering of H_{5}.

In Section 3 we introduce Thas-Walker plane sets of Latin type to get further properties of the second extension of the Thas-Walker construction.

2. The second extension of the Thas-Walker construction

Definition 3. Let Σ be an arbitrary regulization of a spread S of Π. We call

i(Σ) := # ( ⋂_{R∈Σ} R)  (1)

the intersection number of Σ. If i(Σ) = 0, then we say that the regulization Σ is of intersection number 0.

By [6, Remark 2.4], i(Σ) ∈ {0, 1, 2}. From [6, Remark 2.5 and 2.6] we deduce:

Remark 1. A regulization Σ is of intersection number 0 if, and only if, Σ is asymplecticly complemented or unisymplecticly complemented or elliptic.

In the following remark we sum up further properties of a regulization of intersection number 0; for (1) and (2) see [6, Remark 2.8 resp. 2.9], the statements (3) and (4) are evident.

Remark 2. Let S be a spread of Π admitting the regulization Σ of intersection number 0. Then following statements hold true:

(1) Each element of S belongs to exactly one regulus of Σ.

(2) The complementary congruence S_{Σ}^{c} = ∪(R^{c}|R ∈ Σ) of S with respect to Σ is also a spread of Π.

(3) {R^{c}|R ∈ Σ} =: Σ^{c} is a regulization of S_{Σ}^{c} with i(Σ^{c}) = 0.

(4) The complementary congruence (S_{Σ}^{c})_{Σ^{c}}^{c} of S_{Σ}^{c} with respect to Σ^{c} coincides with S; in symbols:

(S_{Σ}^{c})_{Σ^{c}}^{c} = S.  (2)

2In contrast to a linear flock, a linear flocklet is not uniquely determined by the common line of the carrier planes of the participating proper conics.

The rest of this section generalizes [ETW, Section 3].

Proposition 1. Let S be a spread of Π and let Γ be a regulization of S with intersection number 0. Then {λ(R^{c})|R ∈ Γ} =: C is a flocklet of the Klein quadric H_{5}.

Proof. Clearly, (RZ2) implies (FT2). By Remark 2 (3), S_{Γ}^{c} is a spread admitting the regulization Γ^{c} with i(Γ^{c}) = 0; we can say that C is the λ-image of the regulization Γ^{c}. Let η be an arbitrary Latin plane on H_{5}, then λ^{−1}(η) is a star of lines with a vertex, say Y. A conic k of C has a non-empty intersection with η if, and only if, the regulus λ^{−1}(k) of Γ^{c} contains a line incident with Y. In the spread S_{Γ}^{c} there exists exactly one line, say s_{Y}, incident with Y. Because of i(Γ^{c}) = 0 and Remark 2 (1), s_{Y} belongs to exactly one regulus, say R_{Y}^{c}, of Γ^{c}. Now Y ∈ s_{Y} ∈ R_{Y}^{c} ∈ Γ^{c} implies λ(s_{Y}) ∈ η ∩ λ(R_{Y}^{c}) and λ(R_{Y}^{c}) ∈ C, i.e., the conic λ(R_{Y}^{c}) is the only element of C having at least one common point with η. □

Remark 3. Let D be a flockoid of the Lie quadric L_{4} with L_{4} ⊂ H_{5}. Then D is a flocklet and also a flockling of H_{5}.

Proof. Let ξ be an arbitrary (Latin or Greek) plane on H_{5}. Then ξ ∩ span L_{4} is always a line, say x. Because of (FD1)^{3}, there exists exactly one conic k_{x} ∈ D with k_{x} ∩ x ≠ ∅ and thus k_{x} ∩ ξ ≠ ∅. □

By [ETW, Remark 4], each Lie quadric of PG(4, 𝕂) is embeddable into the Klein quadric H_{5} of PG(5, 𝕂), hence each flockoid of a Lie quadric can be interpreted as flocklet and also as flockling of a suitable Klein quadric.

Remark 4. Let F be a flock of the elliptic quadric Q_{e} with Q_{e} ⊂ H_{5}. Then F is a flocklet and also as flockling of H_{5}.

Proof. Let ξ be an arbitrary (Latin or Greek) plane on H_{5}. Then ξ ∩ span Q_{e} is always a point, say X. In the flock F there exists exactly one conic k_{X} with X ∈ k_{X}. □

By [ETW, Remark 9], each elliptic quadric of PG(3, 𝕂) is embeddable into a Lie quadric of PG(4, 𝕂) which in turn is embeddable into the Klein quadric of PG(5, 𝕂), by [ETW, Remark 4]. Hence each elliptic flock can be interpreted as flocklet and also as flockling of a suitable Klein quadric.

Before formulating and proving the converse of Proposition 1 in Proposition 2 we expose two lemmas about flocklets. The statements of the following Lemma 1 are immediate consequences of (FT1) and the properties of a plane section of a quadric.

Lemma 1. Let A be a flocklet of the Klein quadric H_{5}.

(i) Then different conics of A are disjoint. (ii) If {P_{1}} and {P_{2}} are different improper conics of A, then P_{1} ∨ P_{2} ⊄ H_{5}. (iii) If η is a Latin plane on H_{5} and k ∈ A satisfies k ∩ η ≠ ∅, then η ≠ span k, η ∩ span k is no line, and #(k ∩ η) = 1.

Lemma 2. Let A be a flocklet of the Klein quadric H_{5} and let k_{1} be a proper conic of A. If k_{2} ∈ A \ {k_{1}}, then there exists no tangent cone C_{4} of H_{5} with k_{1} ∪ k_{2} ⊂ C_{4}.

3Compare [ETW, Definition 3].

Proof. Let C_{4} be a tangent cone of H_{5} with a vertex, say V, and with k_{1} ⊂ C_{4}. In span C_{4} there exists a 3-dimensional subspace, say S_{3}, with V ∉ S_{3}. Now C_{4} ∩ S_{3} =: Q_{h} is a hyperbolic quadric and the Latin planes on C_{4} give rise to a regulus R_{Q_{h}} on Q_{h}. By k_{1}^{π} we denote the image of k_{1} under the projection from centre V onto S_{3}. The proper conic k_{1}^{π} ⊂ Q_{h} meets each line of the regulus R_{Q_{h}} in exactly one point. Consequently, k_{1} and each Latin plane on C_{4} have exactly one common point and, because of (FT1), C_{4} cannot contain further (proper or improper) conics of A. □

Proposition 2. If A is a flocklet of the Klein quadric H_{5}, then

∪((λ^{−1}(k))^{c}|k ∈ A) =: T_{E_{2}}(A)  (3)

is a spread of Π admitting the regulization

{(λ^{−1}(k))^{c}|k ∈ A} =: T_{R_{2}}(A)  (4)

and T_{R_{2}}(A) is of intersection number 0.

Proof. Let X be an arbitrary point of Π and denote the star of lines with vertex X by L[X]. In T_{E_{2}}(A) there exists a line incident with X if, and only if, there is a conic k_{X} ∈ A such that X is on a line h of the regulus λ^{−1}(k_{X}), i.e., h ∈ L[X]∩λ^{−1}(k_{X}) and thus λ(h) ∈ λ(L[X])∩ k_{X}. As λ(L[X]) is a Latin plane on H_{5}, so there is a unique k_{X} ∈ A with k_{X} ∩ λ(L[X]) ≠ ∅, by (FT1) . Hence there is a unique regulus in T_{E_{2}}(A), namely (λ^{−1}(k_{X}))^{c}, which contains a line incident with X. Consequently, T_{E_{2}}(A) is a spread.

Next we prove the validity of (RZ1) and (RZ2) for T_{R_{2}}(A). Clearly, (FT2) implies (RZ2). Instead of (RZ1) we show even more: (RZ1*) Each line of T_{E_{2}}(A) belongs to exactly one regulus of T_{R_{2}}(A). Let b ∈ T_{E_{2}}(A) be arbitrary. We assume, to the contrary,

b ∈ (λ^{−1}(k_{1}))^{c} ∩ (λ^{−1}(k_{2}))^{c}, {k_{1}, k_{2}} ⊆ A, k_{1} ≠ k_{2}.  (5)

In the case that both (λ^{−1}(k_{1}))^{c} and (λ^{−1}(k_{2}))^{c} are improper reguli with (λ^{−1}(k_{i}))^{c} = {g_{i}} and g_{i} ∈ L, (i = 1, 2), the lines g_{1} and g_{2} are skew and (5) yields the absurdity b ∈ {g_{1}}∩{g_{2}} = ∅. Hence we may assume, without loss of generality, that (λ^{−1}(k_{1}))^{c} is a proper regulus. Each line of (λ^{−1}(k_{1})) ∪ (λ^{−1}(k_{2})) meets b. Thus k_{1} ∪ k_{2} is contained in the tangent cone of H_{5} at the point λ(b), a contradiction to Lemma 2. Thus T_{R_{2}}(A) is a regulization and the validity of (RZ1*) implies i(T_{R_{2}}(A)) = 0. □

The process of gaining a spread from a flocklet via formula (3) is called second extension of the Thas-Walker construction.

We combine Proposition 1 and 2 and get

Corollary 1. To each spread of PG(3, 𝕂) admitting a regulization of intersection number 0 there corresponds a flocklet of the Klein quadric of PG(5, 𝕂), and vice versa.

3. Thas-Walker plane sets of latin type

Let Q be an elliptic or hyperbolic quadric of PG(3, 𝕂) with bijective polarity π_{Q}. A point set T of PG(3, 𝕂) is called a Thas-Walker point set with respect to Q, if {π_{Q}(X) ∩ Q | X ∈ T ∧ π_{Q}(X)∩ Q ≠ ∅} is a flock of Q. Let L_{4} be a Lie quadric of PG(4, 𝕂) with polarity π_{4}. A line set T_{ℓ} of PG(4, 𝕂) is called a Thas-Walker line set with respect to L_{4}, if {π_{4}(X)∩ L_{4} | X ∈ T_{ℓ} ∧ π_{4}(X) ∩ L_{4} ≠ ∅} is a flockoid of L_{4}. In [7, Section 2.2], we considered a Thas-Walker point set T with respect to an elliptic quadric E ⊂ H_{5} and got the λ-image of a spread T by projecting T from the line e = π_{5}(span E), with π_{5} the polarity defined by the Klein quadric H_{5}, into H_{5}, in symbols λ(T ) = ∪ ((X ∨ e) ∩ H_{5}| X ∈ T). In [ETW, Section 4], we took a Thas-Walker line set T_{ℓ} with respect to a Lie quadric L_{4} ⊂ H_{5} and got the λ-image of a spread T_{ℓ} by projecting T_{ℓ} from the point Z = π_{5}(span L_{4}) into the Klein quadric H_{5}, in symbols λ(T_{ℓ}) = ∪ ((x ∨ Z) ∩ H_{5}| x ∈ T_{ℓ}). In the present section we continue this process: we shall take a Thas-Walker plane set T_{La} of Latin type^{4} with respect to the Klein quadric H_{5} and shall get the λ-image of a spread T_{La} by projecting T_{La} from π_{5}(span H_{5}) = ∅ into the Klein quadric H_{5}. As this projection is the identity of P_{5}, so the following statements and formulas concerning Thas-Walker plane sets of Latin type are of simplier appearance as their analogues in the case of Thas-Walker point resp. line sets.

A set T_{La} of planes of Π_{5} is called Thas-Walker plane set of Latin type with respect to H_{5}, if

D_{2}(T_{La}) := {π_{5}(ξ) ∩ H_{5} | ξ ∈ T′_{La}} with T′_{La} := {ξ ∈ T_{La} | π_{5}(ξ) ∩ H_{5} ≠ ∅} (6)

is a flocklet of H_{5}. We put

T_{La}^{p} := {ξ ∈ T_{La} | #(π_{5}(ξ) ∩ H_{5}) > 1}  (7)

for the set of those planes of T_{La} which yield proper conics.

Remark 5. Let {P } ⊂ H_{5} be an improper conic. In the case 𝕂 = ℝ there are infinitely many planes α with π_{5}(α)∩ H_{5} = {P }, since there are infinitely many planes incident with P and contained in π_{5}(P) and intersecting the tangent cone π_{5}(P)∩ H_{5} only in P (compare also the description of a tangent cone of H_{5} in the proof of Lemma 2). In other words, if T_{La_{1}} and T_{La_{2}} are Thas-Walker plane sets of Latin type with respect to H_{5}, then D_{2}(T_{La_{1}}) = D_{2}(T_{La_{2}}) implies T_{La_{1}}^{p} = T_{La_{2}}^{p}, but not T′_{La_{1}} = T′_{La_{2}}.

For the discussion of Thas-Walker line sets in [ETW] we had to assume Char 𝕂 ≠ 2 througout Section 4. Here we need this additional assumption only in the following

Lemma 3. Assume Char 𝕂 ≠ 2. Denote by P_{L}[H_{5}] the set of all Latin planes on the Klein quadric H_{5}, by the way, π_{5}(P_{L}[H_{5}]) = P_{L}[H_{5}]. A set A of planes of Π_{5} is a Thas-Walker plane set of Latin type with respect to H_{5} if, and only if, the following three conditions hold true^{5}:

4This concept has to be defined still.

5In order to have full conformity with the corresponding Lemma 5 from [ETW] we start the numbering of the conditions with 2.

(TWLa2) #(A_{e}) ≤ 2 with A_{e} := {α ∈ A | α ∩ π_{5}(α) ≠ ∅}. (TWLa3) If α_{e} ∈ A_{e}, then #(π_{5}(α_{e}) ∩ H_{5}) = 1. (TWLa4) For each plane ξ ∈ P_{L}[H_{5}] there exists exactly one plane α ∈ A with ξ ∩ α ≠ ∅.

Proof. If the intersection of the plane α ∈ A and the plane π_{5}(α) is empty, then π_{5}(α) ∩ H_{5} is either a proper conic or empty, and conversely^{6}. We define D_{2}(A) according to (6). Now (TWLa2) and (TWLa3) imply that all elements of D_{2}(A) are proper or improper conics and that D_{2}(A) satisfies (FT2), and vice versa. Finally, (TWLa4)⇔(FT1). □

If k ⊂ H_{5} is a proper conic, then (λ^{−1} (k))^{c} = λ^{−1} (π_{5} (span k)). If α is a plane of Π_{5} such that α ∩ H_{5} is an improper conic, say {A}, then, as the reader proves easily, π_{5}(α) ∩ H_{5} = {A} and hence (λ^{−1} ({A}))^{c} = λ^{−1} (π_{5} (α)). Thus we have the subsequent modification of the second extension of the Thas-Walker construction:

Lemma 4. If T_{La} is a Thas-Walker plane set of Latin type with respect to the Klein quadric H_{5}, then

T_{La} := ∪(λ^{−1}(ξ) | ξ ∈ T_{La})  (8)

is a spread of Π admitting the regulization

Θ_{La} := {λ^{−1}(ξ) | ξ ∈ T′_{La}}  (9)

wherein T′_{La} is defined by (6); Θ_{La} is of intersection number 0.

We say that Φ(T′_{La}) := ∪(τ | τ ∈ T′_{La}) is the 3-surface determined by T′_{La} and that each plane τ ∈ T′_{La} is a T′_{La}-generatrix of Φ(T′_{La}). The following lemma is evident.

Lemma 5. Suppose that the conditions (and notations) of Lemma 4 hold. If each proper conic k with k ⊆ Φ(T′_{La}) ∩ H_{5} is contained in a T′_{La}-generatrix of Φ(T′_{La}), then

(1) each proper regulus contained in the spread T_{La} belongs to Θ_{La};

(2) T_{La} admits exactly one regulization, namely Θ_{La}.

Put^{7}

v′ := dim( ⋁_{ξ∈ T′_{La}} ξ) and v^{p} := dim( ⋁_{ξ∈ T_{La}^{p}} ξ).  (10)

(A) If v′ = v^{p} = 5, then T_{La} is an asymplectic ^{8} spread and D_{2}(T_{La}) is an unbundled flocklet.

In order to give a survey of all imaginable cases of v′ = v^{p} we exhibit the subsequent table. The first row of the table represents an abbreviation of statement (A). The first column of the table gives the assumption, middle and left column are the conclusions.

v′ = v^{p}  T_{La}  D_{2}(T_{La})
5  asymplectic unbundled
4  unisymplectic star
3  regular  linear

Table 1. Deductions from v′ = v^{p} 6This assertion is wrong for Char 𝕂 = 2.

7Compare also the definition of a 4-spatial or solid or planar Thas-Walker line set in [9, Definition 1]; furthermore, see [9, Remark 10].

8A spread S of Π is called symplectic, if S is part of a linear complex of lines; otherwise we say that S is an asymplectic spread. By a unisymplectic spread we mean a non-regular symplectic spread.

Definition 4. If a flocklet B of the Klein quadric H_{5} is neither a flockoid of a Lie quadric L_{4} ⊂ H_{5} nor a flock of an elliptic quadric Q_{e} ⊂ H_{5}, then we say that B is a genuine flocklet^{9}.

Put

d′ := dim( ⋂_{ξ∈ T′_{La}} ξ) and d^{p} := dim( ⋂_{ξ∈ T_{La}^{p}} ξ).  (11)

(B) If d′ = d^{p} = −1, then D_{2}(T_{La}) is a genuine flocklet and Θ_{La} is an asymplecticly complemented regulization.

The following table gives a survey.

d′ = d^{p}  Θ_{La}  D_{2}(T_{La})
−1  asymplecticly complemented genuine flocklet
0  unisymplecticly complemented genuine flockoid
1  elliptic  elliptic flock

Table 2. Deductions from d′ = d^{p}

4. Spreads with regulizations of intersection number 0: types

Given a spread S of Π with regulization Σ we can distinguish 9 combinations:

Combination 1: S regular and Σ elliptic: Examples are well-known. Combination 2: S regular and Σ unisymplecticly complemented: See [9, Section 5, Type 7]. Combination 3: S regular and Σ asymplecticly complemented: See the subsequent Remark 6. Combination 4: S unisymplectic and Σ elliptic: See [7, Theorem 3.3.1]. Combination 5: S unisymplectic and Σ unisymplecticly complemented: See [9, Section 5, Type 3, Type 4, Type 5, and Type 6]. The case from [9, Section 5, Type 4] is investigated in [9, Section 6] shortly. Combination 6: S unisymplectic and Σ asymplecticly complemented: See [6, Remark 4.1.2 and (4.1.7)]. Combination 7: S asymplectic and Σ elliptic: See [7, Theorem 3.2.1]. Combination 8: S asymplectic and Σ unisymplecticly complemented: See [9, Section 5, Type 1 and Type 2] which are investigated in [10] thoroughly. Combination 9: S asymplectic and Σ asymplecticly complemented: For sake of completeness, we construct an example in the subsequent Remark 7.

Remark 6. We use the concepts and notations of [6, Section 4]. Let E be a regular spread of PG(3, ℝ). We decompose the elliptic quadric λ(E) by two (proper) disjoint conics c_{1}, c_{2} of λ(E) into two elliptic caps U_{1}, U_{2}, and an elliptic zone V. Put span c_{1} ∩span c_{2} =: v. Choose lines g_{j} (j = 1, 2) with g_{j} ⊂ span c_{j}, g_{j} ∩ c_{j} = ∅, and g_{j} ≠ v such that g_{1} and g_{2} are skew. Then

Φ := λ̃^{−1}(C(U_{1}, g_{1}) ∪ C(V, v) ∪ C(U_{2}, g_{2}))  (12)

is an asymplecticly complemented regulization of E.

9Compare also the definition of a genuine flockoid of a Lie quadric given in [9, Definition 1].

Remark 7. Based on [5, Satz 5] in [6, 4.3] two spreads S_{0} and S_{1} are defined; with the notations from [5, Satz 5] we have:

S_{0} = E_{1}^{1} ∪ R_{1,1}^{r} ∪ E_{2}^{2} ∪ R_{2,1}^{r} ∪ E_{3}^{3}  (13)

wherein R_{1,1}^{r} and R_{2,1}^{r} are proper reguli described in [5, 3.2]; λ(S_{0}) is composed of the elliptic caps λ(E_{1}^{1} ∪ R_{1,1}^{r}) =: W_{1}, λ(R_{2,1}^{r} ∪ E_{3}^{3}) =: W_{2} and the elliptic zone λ(R_{1,1}^{r} ∪ E_{2}^{2} ∪ R_{2,1}^{r}) =: Z with disjoint limiting conics. Consider the three solids

S_{1}^{1} := span λ(W_{1}) = {pℝ ∈ P_{5} | p_{0} + p_{3} = p_{1} − ε p_{2} + p_{4} + ε p_{5} = 0}
S_{2}^{2} := span λ(Z) = {pℝ ∈ P_{5} | p_{0} + p_{3} = p_{1} + p_{4} = 0}  (14)
S_{3}^{3} := span λ(W_{2}) = {pℝ ∈ P_{5} | p_{1} + p_{4} = p_{0} + 4ε′′p_{2} + p_{3} − ε′′p_{5} = 0}

( ε, ε′′∈ ℝ \ \{0\}, |ε| < 1, |ε′′| < \frac{1}{4} );

as dim(S_{1}^{1} ∨ S_{2}^{2} ∨ S_{3}^{3}) = 5, so the spread S_{0} is asymplectic. Choose lines h_{j} (j = 1, 2) with h_{j} ⊂ span λ(R_{j,1}^{r}), h_{j} ∩ λ(R_{j,1}^{r}) = ∅, and h_{j} ≠ z such that h_{1} and h_{2} are skew. Then

Ψ := λ̃^{−1}(C(W_{1}, h_{1}) ∪ C(Z, z) ∪ C(W_{2}, h_{2}))  (15)

is an asymplecticly complemented regulization of S_{0}.

The examples given for the Combinations 7, 8, and 9 show

Corollary 2. There exist genuine linear flocklets, genuine star flocklets, and genuine unbundled flocklets.

The spread S_{0} of Remark 7 is like “patchwork”, therefore we ask for an explicit example of an algebraic asymplectic spread^{10} with asymplecticly complemented regulization. Applying the second extension of the Thas-Walker construction we give in [11] such an example.

I would like to express my thanks to H. Havlicek (Vienna) for valuable suggestions in the preparation of this article.

References

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10A spread S is called algebraic, if S is an algebraic congruence of lines.

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Received September 25, 1999