Printer Friendly

On the existence of projective embeddings of multiveblen configurations.

Introduction

The class of multiveblen configurations was introduced in [11]. The aim of this note is to determine which multiveblen configurations can be embedded into a Desarguesian projective space (comp. [14] (and [5]) for more information on configurations in projective geometry).

The Veblen (or Veblen-Young) configuration is a well known classical (62 43) -configuration of projective geometry (1). Sometimes it is also called a Pasch Configuration, though the original Pasch configuration originates in ordered (Euclidean) geometry, while no geometrical order is involved in the definition of the Veblen configuration. A multiveblen configuration (in short: MVC) is a partial Steiner triple system (i.e. a partial linear space with the lines of size 3, cf. [13]), whose construction generalizes the construction of the Desargues configuration and of the [10.sub.3]G-configuration of Kantor consisting in completing three Veblen configurations on three concurrent lines by a single new line (see Figure 1). Loosely speaking, a multiveblen configuration M = [M.sup.p.sub.n][??]h (shorter: M = [M.sub.n]>h) can be visualized as a system of ([??]) Veblen configurations on n concurrent lines of the size 3 (through a point p) completed by another multiveblen configuration h (of the form h = [M.sub.n-2][??]h', or by some "combinatorial Grassmannian" h), whose lines join "second points of intersection" in the corresponding Veblen configurations. This verbal presentation does not characterize M uniquely; the additional parameter P, a graph on n vertices is used to make the definition correct and we write M = [M.sup.p.sub.n][[??].sub.p]h (cf. 1.1).

The class of multiveblen configurations contains, in particular, structures which generalize the Desargues configuration considered as a perspective of two triangles, and which can be visualized as a perspective of two n-simplices in a projective space. These structures can also be represented in a pure combinatorial way as combinatorial Grassmannians [G.sub.2](n + 2) (cf. [10]).

In a sense, a multiveblen configuration was invented as a solution of a (rather technical) problem to construct and classify sufficiently regular configurations in which any two lines through a given (fixed) point yield a Veblen configuration. This was the idea of constructing structures defined in [11], of the form [M.sup.p.sub.n][[??].sub.p]h with arbitrary h. A huge variety of the obtained structures (2) forced us to restrict ourselves in the paper to the case when h is again a multiveblen configuration, or a combinatorial Grassmannian. In essence, equivalently, we could require that h is a MVC, a point, or a line of size 3. In any case the multiveblen configurations seem interesting on their own, due to their simple and well visualizable internal structure, and close connections with the (classical) Veblen configuration. Since the Veblen axiom is a fundamental axiom of projective geometry, multiveblen configurations can be considered as, loosely speaking, locally projective (for a system of pairwise not collinear points). Then the question which of them are "really" projective i.e. which can be realized in a (Desarguesian) projective space seems natural.

Clearly, both combinatorial Grassmannians and the [10.sub.3]G configuration can be embedded into a projective space. In our note we prove that these are the only simple MVC (i.e. those M where h is a combinatorial Grassmannian) which can be projectively embedded. The problem to characterize all the projectively embeddable MVC is much more complex because the class of all MVC contains configurations of various quite irregular structure. We distinguish the class of regular MVC and prove that in this class, besides combinatorial Grassmannians exactly one new series of projectively embeddable structures appear; configurations in this series generalize the [10.sub.3]G configuration.

1 Notation

First, we briefly recall the definitions of the structures considered in the paper. Let X be a nonempty set and k be an integer; we write [P.sub.k](X) for the family of k-element subsets of X. Two graphs on X are especially important (cf. [16]):

the empty graph [N.sub.X] = (X,[empty set]) and the complete graph [K.sub.X] = (X, [P.sub.2](X)). Frequently, only the type of a graph will be needed; we write [K.sub.n] for the type of [K.sub.X] where [absolute value of X] = n, and similarly [N.sub.n] for the type of [N.sub.X]. In what follows we shall also frequently identify a set P [subset] [P.sub.2](X) with the graph (X, P). If P [subset] [P.sub.2](X) and Y [subset] X we write P [??] Y for the graph P [intersection] [P.sub.2](Y). Given an ordering [x.sub.1], ..., [x.sub.n] of the elements of X we write

[L.sub.n] for the type of the linear graph (X,{[x.sub.1], [x.sub.2]}, {[x.sub.2],[x.sub.3]}, ..., {[x.sub.n- 1],[x.sub.n]}}).

The structure

[G.sub.2](X) := ([P.sub.2](X), [P.sub.3](X), [subset]) [congruent to] ([P.sub.2](X), {[P.sub.2](Y): Y [member of] [P.sub.3](X)}, [member of]) is referred to as a combinatorial Grassmannian (cf. [10], [11], [9]) (3). Then [G.sub.2]([absolute value of X]) is the type of [G.sub.2](X) - it is a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-configuration (n = [absolute value of X]; generally, a ([v.sub.r][b.sub.k]) -configuration is a configuration with v points of degree r each and b lines of size k each), so it is a partial Steiner triple system. In particular, [G.sub.2](5) is the Desargues configuration (cf. [7]).

Construction 1.1. Let h = ([P.sub.2](X),L) be a partial Steiner triple system and P be a non-oriented graph without loops defined on X. We take any two distinct elements [p.sub.1], [p.sub.2] [not member of] X and put p = {[p.sub.1], [p.sub.2]}, X' = X [union] p. Consider the following families of blocks:

[L.sub.1] = {{{[p.sub.1], [p.sub.2]}, {[p.sub.1], i),{[p.sub.2],i}} : i [member of] X},

[L.sub.2] = {{{i,j}, {[p.sub.1],i}, {[p.sub.2],j}},: i,j [member of] X, i [not equal to] j, {i,j} [not member of] P),

[L.sub.3] = {{{i,j}, {[p.sub.1],i}, {[p.sub.1],j}, {i,j},{[p.sub.2],i},{[p.sub.2],j}}: i,j [member of] X, {i,j} [member of] P).

The structure ([P.sub.2] (X'), L [union] [L.sub.1] [union] [L.sub.2] [union] [union] [L.sub.3]) will be denoted by [M.sup.p.sub.X][[??].sub.P]h and it will be called the multiveblen configuration with center p, consistency graph P defined on X, and axial configuration h. A multiveblen configuration is simple if it has a combinatorial Grassmannian as its axial configuration.

A particular role is played in the sequel by the structure

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We write B(n) := B(X), where [absolute value of X] = n, for short.

It is easy to note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The structure [D.sup.o] := B(3) is the [10.sub.3]G-configuration of Kantor (cf. [6]); in the paper this one will also be called the Veronese configuration. (4) The line [G.sub.2](X) of [D.sup.o] = B(X) is referred to as the axis of [D.sup.o].

[FIGURE 1 OMITTED]

Fact 1.2. If h is a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-configuration then [M.sup.p.sub.X][[??].sub.P ]h is a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-configuration.

The construction of the structure [M.sup.p.sub.X][[??].sub.P]h can be visualized in a more geometrical vein, which is more convenient in the analysis of the obtained configurations. Let us adopt the notation of 1.1. Next, write

[a.sub.i] = {[p.sub.1], i}, [b.sub.i] = {[p.sub.2], i} for i [member of] X

and

[c.sub.z] = z for z [member of] [P.sub.2](X), C = {[c.sub.z]: z [member of] [P.sub.2](X)}.

Step A The set p is an arbitrary "abstract new point".

Step B Through p we have the lines [L.sub.i], and the points [a.sub.i], [b.sub.i] on [L.sub.i], for every i [member of] X.

Step C We have a subset P of [P.sub.2](X) distinguished, and after that

if {i,j} [member of] P: we draw lines [A.sub.i,j] = [bar.[a.sub.i],[a.sub.j]] and [B.sub.i,j] = [bar.[b.sub.i],[b.sub.j]]; the point [c.sub.{i,j}] is common for [A.sub.i,j] and [B.sub.i,j],

if {i, j} [member of] [P.sub.2](X) \ P: we draw lines [G.sub.i,j] = [bar.[a.sub.i],[b.sub.j]]; the point [c.sub.{i,j}] is common for [G.sub.i,j] and [G.sub.j,i],

for every {i, j} [member of] [P.sub.2](X). It is seen that the point p and the points [a.sub.i], [b.sub.i] (i [member of] X) have degree n, while (up to now) [c.sub.z] with z [member of] [P.sub.2](X) has degree 2. Moreover, the number of the points [c.sub.z] is ([??]).

The quadruple of lines ([L.sub.i], [L.sub.j], [A.sub.i,j], [B.sub.i,j]) (([L.sub.i], [L.sub.j], [G.sub.i,j], [G.sub.j,i]) resp.) with any distinct i,j [member of] X yields a classical Veblen configuration.

Step D Let h be any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-configuration. Finally, we identify the points [c.sub.z] constructed above with points of h (under some bijection [gamma]) and, consequently, we group the points [c.sub.z] into ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) new lines obtained as coimages of the lines of h under [gamma].

The resulting configuration will be written as [m.sup.p.sub.X][[??].sup.[gamma].sub.p]h. If the point set of h is [P.sub.2](X) it is natural to put [gamma]: [c.sub.{i,j}] [right arrow] {i,j}; comparing with 1.1 we see that [m.sup.p.sub.X][[??].sup.[gamma].sub.P]h [congruent to] [M.sup.p.sub.X][[??].sub.P]h.

The above interpretation justifies the term multiveblen used to name structures of the form [M.sup.p.sub.X][[??].sub.P]h.

A multiplied multiveblen configuration (more precisely: a multiplied simple multiveblen configuration) is any structure of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [p.sup.j] are two-element sets, the set [X.sub.0] and the [p.sup.j] are pairwise disjoint, [X.sub.j] = [X.sub.j_1] [union] [p.sup.j] for j = 1, ..., k, and [P.sub.j] is a graph defined on [X.sub.j] for j = 0, ..., k - 1.

In most parts we shall consider a "standard" representation of the structure defined by (1) taking [X.sub.0] := {1, ..., m} (m [greater than or equal to] 2) and [p.sup.j] := {m + 2j, m + 2j - 1}.

The structure m of the form (1) is a simple multiveblen configuration if k = 1.

Let P', P" be two graphs on a set X. We write P' [approximately equal to] P" if and only if there is a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of maps ([X.sub.1], ..., [x.sub.s] [member of] X) such that the composition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] maps P' onto P" and every [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] switches connections with the vertex [x.sub.j]: [x.sub.j], y are connected in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].(P) if and only if they are not connected in P, for an arbitrary graph P and y [member of] X.

Fact 1.3 ((cf. [11, Prop. 9])). If h is any partial Steiner triple system defined on the set [P.sub.2](X), p is a two-element set disjoint with X, and P', P" are graphs defined on X then P' [equivalent to] P" yields [M.sup.p.sub.X][[??].sub.p']h [congruent to] [M.sup.p.sub.X][[??].sub.p"]h.

A suitable converse variant of 1.3 is also provable:

Theorem 1.4. Let [absolute value of X] [greater than or equal to] 5. f there is an isomorphism of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which maps [p.sup.1] onto [p.sup.2] then [h.sub.1] [congruent to] [h.sub.2] and then there is a graph [P.sub.3] such that [P.sub.1] [approximately equal to] [P.sub.3] [congruent to] [P.sub.2]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [G.sub.2](X) [congruent to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [G.sub.2](X) then there is a graph [P.sub.3] such that [P.sub.1] [approximately equal to] [P.sub.3] [congruent to] [P.sub.2].

A detailed classification of the simple multiveblen configurations [M.sup.p.sub.X][[??].sub.P][G.sub.2] (X) with [absolute value of X] [less than or equal to] 5 is presented in [11]. One result of that investigations will also be used here:

Fact 1.5. Let P be a graph on a set Y and X [member of] [P.sub.4] (Y). Then P [??] X [approximately equal to] [N.sub.4], P [??] X [approximately equal to] [L.sub.4]. or P X X w L4.

2 Projective embeddings

In what follows by a projective embedding of a configuration k = (S, L) or an embedding of k into a projective space B we mean an injective map which associates with the elements of S points of B and with the elements of L lines of B and which preserves (in both directions) the incidence (comp. [4]). As a rule, in the sequel we consider only embeddings into Desarguesian spaces.

Clearly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [G.sub.2](n - 2) has a standard projective embedding into PG(n, 2) (much interesting information on projective embeddings of the structures [G.sub.2](n) can also be found in [2]).

Now, we continue some remarks concerning projective embeddings of the structure [B.sup.o] = B(3) given in [9].

Fact 2.1 ([9, Prop. 5.3]). If [B.sup.o] is embedded into a projective space B then [B.sup.o] lies on a plane of B.

Lemma 2.2. Let us assume that the configuration [B.sup.o] is embedded into a projective space. Then the lines [a.sub.i], [a.sub.j] and [b.sub.i], [b.sub.j] meet on the axis of [B.sup.o] for every 1 [less than or equal to] i < j [less than or equal to] 3.

Proof. From [9, Prop. 5.3], [B.sup.o] lies on a projective plane n and one of the following holds:

a) There are lines A, B of n such that [a.sub.1], [a.sub.2], [a.sub.3] lie on A and [b.sub.1], [b.sub.2], [b.sub.3] lie on B.

b) There is a conic S in n such that the [a.sub.i] and the [b.sub.j] lie on S.

In the corresponding cases we prove our claim as follows:

a) Let {i, j,l} = {1,2,3}. From the Desargues theorem applied to the triangles ([a.sub.i],[b.sub.l],[a.sub.j]) and ([b.sub.i],[a.sub.l],[b.sub.j]) we infer that the lines [a.sub.i], [a.sub.j],[b.sub.i][b.sub.j] meet on the line [c.sub.{i,l}], [c.sub.{l,j}], which is the axis of [B.sup.o].

b) The point p is the pole of the axis L of [B.sup.o] conjugated under S i.e. it is the center of the harmonic homology f with axis L, which leaves S invariant. Since f interchanges the lines [a.sub.i], [a.sub.j] and [b.sub.i], [b.sub.j], these lines meet on the axis of f, as required.

Proposition 2.3. Let P be a graph on n-element set X, let Y [member of] [P.sub.4](X) such that P [??] Y [approximately equal to] [L.sub.4] or P [??] Y [less than or equal to] [N.sub.4]. Then there is no projective embedding of the structure [M.sup.p.sub.X][[??].sub.P][G.sub.2](X) (in short: of [M.sub.n][[??].sub.P][G.sub.2](n)). (5)

Proof. It suffices to prove that there is no projective embedding of the structure B := [M.sup.p.sub.{1,2,3,4}][[??].sub.P'][G.sub.2]({1,2,3,4}), where P' = [L.sub.4] = {{1,2},{2,3},{3,4}} or P' = [N.sub.{1,2,3,4}]. It is seen that B contains two [B.sup.o]-configurations [V.sub.1], [V.sub.2] with the common center p spanned by the lines [L.sub.2], [L.sub.3], [L.sub.1] and [L.sub.2], [L.sub.3], [L.sub.4] respectively.

Suppose that B is embedded into a projective space. From 2.1, [V.sub.1], [V.sub.2] lie in corresponding planes [[PI].sub.1], [[PI].sub.2] which now have two distinct lines [L.sub.2], [L.sub.3] in common; thus [[PI].sub.1] = [[PI].sub.2] =: [PI]. Let [M.sub.i] be the axis of [V.sub.i]. From the definition, [M.sub.i] passes through [c.sub.{2,3}] and from 2.2 we get that [M.sub.i] passes through the common point of the lines [a.sub.2], [a.sub.3], [b.sub.2], [b.sub.3]. Consequently, [M.sub.1] = [M.sub.2] which does not hold in B.

The following technical lemma will be useful in the sequel.

Lemma 2.4. Let P be the graph on the n-element set X such that P [??] A [approximately equal to] [K.sub.3] for every A [member of] [P.sub.3](X). Then P [approximately equal to] [K.sub.n] and [M.sub.n][[??].sub.P][G.sub.2](n) [congruent to] [G.sub.2] (n + 2).

Proof. Suppose that P = [empty set]; then P [??] A [approximately equal to] [N.sub.3] which contradicts assumptions. Thus there is an edge (say e = {1,2}) of P. For arbitrary i [member of] X \ e, since P [??](e [union] {i}) [approximately equal to] [K.sub.3], either {1, i}, {2, i} [member of] P or {1, i}, {2, i} [not equal to] P. Let us set

[X.sup.+] := {i: {1,i} [member of] P} and [X.sup.+] := {i: {1,i} [not member of] P}.

Observing triples {i, j, 1} with i, j [member of] X \ e we get

i, j [member of] [X.sup.+] or i, j [member of] [X.sup.-] [??] {i, j} [member of] P

i [member of] [X.sup.+], j [member of] [X.sup.-] [??] {i, j} [not equal to] P.

The composition of all the maps [[mu].sub.i] with i [member of] [X.sup.-] transforms P onto [K.sub.n].

Now we are in a position to prove a first important result:

Theorem 2.5. Let P be an arbitrary graph on n (n [greater than or equal to] 4) vertices such that B := [M.sub.n][[??].sub.P][G.sub.2](n) [??] [G.sub.2](n + 2). Then B cannot be embedded into a projective space.

Proof. Let X = {1, ..., n} be the set of the vertices of P. From the assumption and 2.4, there is A [member of] [P.sub.3](X) such that [P.sub.0] := P [??] A [not approximately equal to] [K.sub.3] and thus [P.sub.0] [approximately equal to] [N.sub.3] . Without loss of generality we can assume that A = {1,2,3} and

[P.sub.0] = {{1,2}, {1,3}} or [P.sub.0] = [empty set].

Applying [[mu].sub.1], if necessary, we assume that [P.sub.0] = {{1,2}, {1,3}}.

Observe A' = A [union] {4} and P' = P [??] A'. Note that a graph equivalent to [K.sub.4] defined on A' is one of the following: [K.sub.4], a triangle [C.sub.3] embedded into A', or consisting of two disjoint edges. Restriction of no one of them is the path [P.sub.0] and thus P' [approximately equal to] [N.sub.4] or P' [approximately equal to] [L.sub.4]. In any case from 2.3 and 2.3 we get that the substructure

[M.sub.4][[??].sub.P'][G.sub.2](A') of 95 cannot be projectively embedded.

The result of 2.5 can also be read as follows:

Corollary 2.6. Let B be a simple multiveblen configuration with point degree at least 4. Then B can be embedded into a projective space if and only if B is a generalized Desargues configuration (a combinatorial Grassmannian).

A direct analogue of 2.6 for multiplied multiveblen configurations does not hold.

Proposition 2.7. The structure

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

can be embedded into a Desarguesian projective space if and only if V is equivalent to the linear graph {{1,2}, {1,3}, {2,4}}. If M is embedded into a projective space then [a.sub.3],[a.sub.4] and [bar.[b.sub.3],[b.sub.4]] pass through [c.sub.{1,2}] as well.

Proof. From the definition, the following triples

[V.sub.1] := {{3,4}, {1,3}, {1,4}}, [V.sub.2] := {{3,4}, {2,3}, {2,4}},

[V.sub.3] := {{1,2}, {1,3}, {2,4}}, [V.sub.4] := {{1,2}, {1,4}, {2,3}}.

are the lines of the Veblen configuration B(2).

Let [D.sub.i] be the restriction of M spanned by the lines [L.sub.1], [L.sub.3], [L.sub.4] for i = 1,2. Then [D.sub.i] is either a Desargues or a Veronese [D.sup.o] configuration with the axis [V.sub.i].

Let us assume that M is embedded into a projective Desarguesian space B. For every two distinct [i.sub.1], [i.sub.2] in {1,2,3,4} we have in M a Veblen configuration inscribed into the lines [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the point of intersection of the corresponding lines of M. Let us denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the intersection point (considered in B) of the lines

- [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when {[i.sub.1], [i.sub.2]} [member of] P and of

- [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when {[i.sub.1], [i.sub.2]} [not member of] P.

The point [c.sub.{12}] is determined by the structure of B (2) as the intersection point of the lines [bar.[c.sub.{1,3}, C.sub.{2,4}] = [V.sub.3] and [C.sub.{i,4}]/[C.sub.{2,3}] = [V.sub.4]. In any case two possibilities arise:

(a) [c.sub.{1,2}] | [[bar.a.sub.1],[a.sub.2]], [[bar.b.sub.1],[b.sub.2]] (i.e. {1,2} [member of] P), or

(b) [c.sub.{1,2}] | [[bar.a.sub.1],[b.sub.2]], [[bar.b.sub.1],[a.sub.2]] (i.e. {1,2} [not member of] P).

Step 1: Assume, first, that [D.sub.1] and [D.sub.2] both are Desargues configurations. Without loss of generality we can assume that the given Desargues configurations represent a perspective of the triangles ([a.sub.1], [b.sub.3], [a.sub.4]), ([b.sub.1], [a.sub.3], [b.sub.4])--in [D.sub.1] and ([a.sub.2], [b.sub.4], [a.sub.3]), ([b.sub.2], [a.sub.4], [b.sub.3])--in [D.sub.2]. Consequently, in this case

{1,4}, {2,3} are in P, {1,3}, {3,4}, {2,4} are not in P.

Let us analyze the two possibilities (a) and (b).

Ad (a): Applying several times the Desargues axiom we get [c.sup.*.sub.{3,4}], [c.sub.{1,2}] | [V.sub.3], [V.sub.4] and thus the equality [c.sup.*.sub.{3,4}] = [c.sub.{1,2}] must hold.

Ad (b): From the Desargues axiom applied to the triangles [a.sub.1], [b.sub.2], [b.sub.3] and [b.sub.1], [a.sub.2], [a.sub.3] we obtain L [(c.sub.{1,2}], [c.sub.{1,3}, [C.sub.{2,3})], which gives [C.sub.{2,3}] | [V.sub.3]. Thus [c.sub.{1,2}], [C.sub.{2,3}] | [V.sub.3], [V.sub.4] yields, contradictory, [V.sub.3] = [V.sub.4].

Step 2: Next, assume that [D.sub.1] is a Desargues configuration and [D.sub.2] is a Veronese configuration. Without loss of generality we can label the points on the lines [L.sub.i] in such a way that [D.sub.1] represents the perspective of the triangles ([a.sub.1], [a.sub.3], [b.sub.4]) and ([b.sub.1], [b.sub.3], [a.sub.4]), and [D.sub.2] contains the closed hexagon ([a.sub.2], [b.sub.4], [a.sub.3], [b.sub.2 , [a.sub.4], [b.sub.3]). In this case

{1,3} is in P, {2,3}, {3,4}, {2,4}, {1,4} are not in P.

Let us analyze the two possibilities (a) and (b).

Ad (a): Note that the points [a.sub.1], [a.sub.2], [b.sub.4] are not collinear (otherwise an extra incidence [b.sub.4] | [bar.[[a.sub.1],[a.sub.2]] holds in M) and, analogously, [b.sub.1], [b.sub.2], [a.sub.4] are not collinear. From the Desargues axiom we have L[(c.sub.{1,2}]/[C.sub.{2,4}, [C.sub.{1,4}), so [c.sub.{2,4}]. | [V.sub.4] which gives [V.sub.3] = [V.sub.4].

Ad (b): Since [b.sub.2] | [bar.[[a.sub.1],[a.sub.3]] the points [a.sub.1], [a.sub.3], [b.sub.2] are not collinear. From the Desargues axiom we infer that L[(c.sub.{1,3}] [c.sub.{2,3}], [c.sub.{1,2}); then [c.sub.{2,3}] | [V.sub.3] and thus [V.sub.3] = [V.sub.4].

Step 3: Finally, let us assume that [D.sub.1] and [D.sub.2] both are Veronese configurations. From 2.2 we get that [C.sub.{3,4}], [c.sup.*.sub.{3,4}] | [V.sub.1], [V.sub.2], which yields a contradiction.

Therefore, only in the case analyzed in Step 1 we can expect that M can be embedded into a projective space. On the other hand, let B be the projective 3-space over a field with characteristic [not equal to] 2 and let [beta] be a scalar with [beta] [not equal to] 1, -1. It is a matter of a simple computation that the following map embeds M into P:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

provided that P consists of the edges {1,2}, {1,4}, and {2,3}.

Let [R.sub.4] be the projectively embeddable ([15.sub.4] [20.sub.3]) -multiveblen configuration constructed in the proof of 2.7. It is evident that R4 is not isomorphic to G2 (6), the second projectively embeddable ([15.sub.4] [20.sub.3])-multiveblen configuration. A picture of [R.sub.4] is presented in Figure 2.

Generally, investigations on general (iterated) multiveblen configurations are much more complex. The main reason is that a representation of a multiveblen configuration M in the form (1) is not unique in the sense that M does not determine k, nor [absolute value of [X.sub.0]], nor the [P.sub.i]. Let us point out three simple examples. Let p, q, r be pairwise disjoint two-element sets.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

- k in (1) is not determined by the configuration M.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(cf. [11, Prop. 18])--a multiveblen configuration [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined in 1.1 does not determine its consistency graph P and its axial configuration H.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

even though [N.sub.p] [approximately equal to] [K.sub.p]. This yields, in particular, that even if [P'.sub.0] [approximately equal to] [P".sub.0] the two structures [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] may stay nonisomorphic (comp. 1.3).

Let us say that a multiveblen configuration M is regular if and only if it can be represented in the form (1), where [P.sub.i-1] = [P.sub.i] [??] [X.sub.i-1] for i = 1, ..., k - 1. To determine possible projective embeddings of regular multiveblen configurations we use intensively the following lemma, which follows from 2.7 and 2.3.

Lemma 2.8. Let M = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [absolute value of X] [greater than or equal to] 2, p,q are two-element sets such that X, p, q are pairwise disjoint, [P.sub.1] is a graph on X [union] p, and [P.sub.0] = [P.sub.1] [??] X. Assume that M has a projective embedding. Then the following holds:

We see that while [G.sub.2] (6) represents a perspective of two tetrahedrons (cf. [10]), the configuration [R.sub.4] also represents a perspective, a perspective of some kind of two 4-tuples ([a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4]) and ([b.sub.1], [b.sub.2], [b.sub.3], [b.sub.4]). A perspective of the same type can be seen with the point [c.sub.{3,4}] as the perspective center. Note also that the Desargues axiom applied to this configuration forces the line which joins intersection points of the lines in pairs [bar.([[a.sub.1], [a.sub.3]], [bar.[[a.sub.2], [a.sub.4]]) and [[bar.([b.sub.1], [b.sub.3]], [bar.[b.sub.2], [b.sub.4]]) to pass through p.

[FIGURE 2 OMITTED]

(i) [absolute value of [P.sub.2](X) \ [P.sub.0]] [greater than or equal to] < 1 (i.e. either [P.sub.0] is the complete graph [K.sub.X] or it is [K.sub.X] with exactly one edge deleted).

(ii) If [absolute value of [P.sub.2](X) \ [P.sub.0]] = 1 then [absolute value of X] [less than or equal to] 3.

(iii) One of the following three conditions holds:

1. [P.sub.1] contains p and every pair {s, t} with s [member of] p and t [member of] X,

2. [P.sub.1] contains p and {s, t} [not member of] [P.sub.1] for every s [member of] p, t [member of] X,

3. p [not member of] [P.sub.1], {[i.sub.2], t} [not member of] [P.sub.1] and {[i.sub.1],t} [member of] [P.sub.1] for every t [member of] X, where p = {[i.sub.1], [i.sub.2]}.

In every one of the above three cases either [P.sub.1] [approximately equal to] [K.sub.X[union]p] or [P.sub.1] is equivalent to [K.sub.X[union]p] with one edge (taken from [P.sub.2] (X)) deleted.

Keeping in mind the equality [G.sub.2](X [union] p) [??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [G.sub.2] (X) we see that every combinatorial Grassmannian [G.sub.2] (X) can be presented in the form (1), where either [absolute value of [X.sub.0]] = 2 and then the series (1) begins with a single point [G.sub.2]([X.sub.0]) = [G.sub.2](2), or [absolute value of [X.sub.0]] = 3 and then (1) begins with a single line [G.sub.2] ([X.sub.0]) = [G.sub.2] (3).

Let n [greater than or equal to] 2 be an integer. We define the regular multiveblen configuration [R.sub.n] as follows. If n = 2k for some integer k then we take m = 2, if n = 2k + 1 we set m = 3; in both cases we obtain n = m + 2(k - 1). Let q = {1,2}, X = {1, ..., m}, [p.sup.j] = {m + 2j _ 1, m + 2j} for j = 1, ..., k; in particular, [p.sup.k] = {m + 2k - 1, m + 2k}. Let P = [P.sub.2](X [union] [p.sup.1] [union] ... [union] [p.sup.k-1]) \ {q}. If m = 2 then [P.sub.0] := P [??] X = [N.sub.X]; if m = 3 then [P.sub.0] [approximately equal to] [N.sub.X]. Let us put [P.sub.j] = P [??](X [union] [p.sup.1] [union] ... [union] [p.sup.j]) for 0 < j < k. We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In particular, [R.sup.2] is simply B (q), [R.sub.3] [??] B (3) is the Veronese configuration [V.sup.o], and (up to an isomorphism) [R.sub.4] is the configuration on Figure 2, defined in the proof of 2.7. In general, [R.sub.n] is a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] -configuration. Intuitively, we can write [R.sub.n+2] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Construction 2.9. Let X = {1,2,3}, r = {1,2}, [p.sup.1] = p = {4,5}, [p.sup.2] = q = {6,7}, and let [P.sub.1] = [P.sub.2] (X [union] p) \ r be a graph on X [union] p; let [P.sub.0] = [P.sub.1] [??] X. Evidently [P.sub.0] w [N.sub.X] and thus [M.sub.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a Veronese configuration. By definition, [R.sub.5] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Next, let p be the projective 3-space over a field with characteristic unequal to two and let [beta], [gamma] be two scalars such that [gamma], [beta] [not equal to] 1, -1,0 and [gamma] [not equal to] 2, -2. Let us consider the following map F defined on the points of [R.sub.5]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

A direct though tedious computation shows that the map F embeds [R.sub.5] into B. Thus our construction proves that the ([21.sub.5] [35.sub.3]) -multiveblen configuration [R.sub.5] can be embedded into a projective space.

The above embedding can be further extended to an embedding of R7 (its soundness was verified with the help of Maple V, [rho] [not equal to] 1,-1,2, -2, u/v, v/u, [beta], [beta][rho] [not equal to] 1, and similar); we put:

F([a.sub.7]) = [1, [w.sub.1], [w.sub.2], [w.sub.3]], F([b.sub.7]) = [1, [rho][w.sub.1], [rho][w.sub.2], [rho][w.sub.3],

F([b.sub.6]) = [1, [w.sub.1], - [w.sub.2], [w.sub.3]], F([a.sub.6]) = [1, [rho][w.sub.1], - [rho][w.sub.2], [rho][w.sub.3]].

After that the coordinates of the points F[(c.sub.{i,j}]) with i [member of] {6,7} can be directly computed.

We see that the above procedure can be continued by adding suitable pairs [a.sub.8], [b.sub.8], [a.sub.9], [b.sub.9], ... [a.sub.2k], [b.sub.2k], [a.sub.2k+1], [b.sub.2k+1] on lines through q, which should yield an embedding of [R.sub.2k+1] into B.

Construction 2.10. Now, we shall extend the embedding of [R.sub.4] given in the proof of 2.7 to a projective embedding of [R.sub.6]. First, we note that the substructure of [R.sub.5] spanned by the points on the lines through q and [a.sub.1], [a.sub.2], [a.sub.4], [a.sub.5] is exactly [R.sub.4]; its embedding (2) given in the proof of 2.7 differs from that of 2.9 in the ordering of some symbols in pairs [a.sub.i], [b.sub.i].

Let F be defined as in 2.9. Let us write [sigma]i = i for for i [less than or equal to] 2 and [sigma]i = i + 1 for 2 < i [less than or equal to] 6 and let us label the points on the lines of [R.sub.6] through [p.sup.3] = F(q) by [a.sub.i], [b.sub.i] in a standard way. It is a matter of simple (computer-aided) computation that the map G defined by G([a.sub.i]) = F([a.sub.[sigma]i]), G([b.sub.i]) = [F(b.sub.[sigma]i), [G.sub.(c{i,j})] = F(c.sub.{[sigma]i, [sigma]j}) is an embedding of [R.sub.6] into B.

We see that this embedding can be further extended to an embedding of [R.sub.2k] to [beta].

Finally we obtain our main result

Theorem 2.11. Let M be a regular multiveblen configuration. If M can be embedded into a projective space then either M [??] [G.sub.2] (n) or M [??] [R.sub.n] for some integer n.

Proof. Let M be defined by (1); we define inductively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and then M = [M.sub.k]. Recall that the structure [M.sub.j] is defined on [p.sub.2] ([X.sub.j]). Let m = [absolute value of [X.sub.0]] and let n [greater than or equal to] 4 be the degree of a point in M. From the construction, [absolute value of [X.sub.k]] = m + 2k; we have n = (m + 2k) - 2 and thus [absolute value of [X.sub.k]] = n + 2. From the assumptions, the graph [P.sub.k-1] determines all the graphs [P.sub.j] with j < k - 1.

Clearly, if [M.sub.k] can be projectively embedded, its subspace [M.sub.k-1] can be projectively embedded as well; continuing we obtain that Mj can be projectively embedded for every j = 0, ..., k - 1.

If k = 1 then for m < 4 our claim is evident: M is either the Veblen configuration [G.sub.2](4) (m = 2), or the Desargues configuration [G.sub.2] (5) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [G.sub.2]([X.sub.0]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or the Veronese configuration [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If m [greater than or equal to] 4 from 2.5 we get that M admits a projective embedding if and only if M [??] [G.sub.2]([X.sub.0] [union] p) for some two-element set p.

If k > 1 we apply consecutively 2.8 to determine possible [P.sub.j]

for j = 0, ..., k - 2. Assume that there are distinct [i.sub.0], [j.sub.0] [member of] [X.sub.0] with X' := {[i.sub.0], [j.sub.0]} [not member of] [P.sub.0]. In view of 2.8 we have m = [absolute value of [X.sub.0]] [less than or equal to] 3; from 2.7, 2.8, and 2.9 we infer that either [M.sub.2] is [R.sub.4] (m = 2, [M.sub.1] [??] B(2)) or it is [R.sub.5] (m = 3, [M.sub.1] [??] [V.sup.o]). In both cases X' [not member of] [P.sub.l] for l > 0 but [absolute value of [X.sub.l]] > 3 so, in view of 2.8, the graph [P.sub.l] has the form [p.sub.2] ([X.sub.l]) \ X' and [M.sub.l] = [R.sub.m+2l]. (In accordance with 2.9 and 2.10 structures [M.sub.l] with l > 2 can be projectively embedded and after all M can be projectively embedded.)

Finally, suppose that there is a pair s := {i, j} of distinct elements of [X.sub.l] such that s [member of] [P.sub.l] for some l > 0 but [P.sub.l], = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This yields, in particular, that [M.sub.l'+1] = [G.sub.2]([X.sub.l']) for every l' < l and thus [M.sub.l] = [G.sub.2]([X.sub.l-1]). If l = k - 1 from 2.5 we obtain that necessarily [M.sub.k] [??] [G.sub.2] ([X.sub.l]). Assume that l < k - 1 and have a look at [M.sub.l+1]. Applying 2.8 we obtain [absolute value of [X.sub.l]] [less than or equal to] 3, which is impossible and thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A Starting from another representation of the Veblen configuration

The construction of a multiveblen configuration as defined in (1) can also begin with the representation of the Veblen configuration H in the form H = [G.sup.*.sub.2]; (4) = ([p.sub.2](X), [p.sub.1](X), [contains]>) (where [absolute value of X] = 4, cf. [10]). To complete our results we prove

Proposition A.1. Neither [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be embedded into a Desarguesian projective space.

Proof. Let X = {1,2,3,4} and P be a graph defined on X such that P [approximately equal to] [K.sub.4] or P [approximately equal to] [L.sub.4]. Say, P = [K.sub.x] or P = {{1,2}, {2,3}, {3,4}}. Consider M = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose that M is embedded into a Desarguesian projective space B. In any case P [??] Y [approximately equal to] [K.sub.Y], where Y = {1, 2, 4}. Consequently, {1, 2}, {1, 4} and {2, 4} are collinear in B. From the definition of [G.sup.*.sub.2], (X) the points {1,2}, {1,4}, {1,3} are collinear as well and thus all the points of G2; (X) should lie on one line of B, which is impossible.

Proposition A.2. The structure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (4) can be embedded into a Desarguesian projective space. (6)

Moreover, when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4) is embedded into a Desarguesian projective space B then the characteristic of the coordinate field of B is 2, that is p satisfies the projective Fano axiom.

Proof. Write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where X = {1,2,3,4}. Let us start with an analysis of possible embeddings of [R.sup.*.sub.4] into a projective space B. As usual we write [c.sup.*.sub.{i,j}] for the common point of [bar.[a.sub.i], [a.sub.j]], [bar.[b.sub.i], [b.sub.j]] which exists in B. Observing the triangles [a.sub.1], [a.sub.2], [b.sub.3] and [b.sub.1], [b.sub.2], [a.sub.3] of B with the perspective center p we obtain that the points [c.sub.{13}], [c.sub.{2 3}] and [c.sup.*.sub.{1,2}] are collinear; similarly, the points [c.sub.{1,4}], [c.sub.{2,4}]}, [c.sup.*.sub.{1,2}] are collinear, which gives [c.sup.*.sub.{1,2}] = [c.sup.*.sub.{3,4}]. With the same technique we obtain

[c.sub.{i,j}] = [c.sup.*.sub.i',j'}] whenever {i, j, i', j'} = X. (4)

Therefore, the given points yield an embedding of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [G.sub.2] (X) as well - it suffices to note that with k(u) = X \ u for u [member of] P(X) we obtain: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a line of G22 (X) if and only if c^, cu2, is a line of G2(X). And conversely, every projective embedding of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [G.sub.2] (X) which satisfies (4) yields a projective embedding of [R.sup.*.sub.4].

Next, since [C.sub.{3;4}] | [bar.[a.sub.1], [a.sub.2]], [bar.[a.sub.3], [b.sub.4]], [C.sub.{2,4}] | [bar.[a.sub.2], [b.sub.4]], [bar.[a.sub.1], [a.sub.3]], and [c.sub.{1,4}] | [bar.[a.sub.1],[b.sub.4]], [bar.[a.sub.2],[a.sub.3]] the points [a.sub.1], [a.sub.2], [a.sub.3], [b.sub.4] yield in p a quadrangle with the diagonal points [c.sub.{2,4}], [c.sub.{2,4}] [c.sub.{3,4}], which are collinear and thus B contains a closed Fano configuration.

Finally, to construct an embedding of [R.sup.*.sub.4] into a Desarguesian projective space it suffices to take a closed Fano configuration in a projective space B (thus coordinatized by a iield with characteristic 2; to ensure that the procedure works one can assume that dim([beta]) > 2), a point p not on a plane that contains this configuration, and an image of this Fano configuration under suitable homology.

However, no series of projectively embeddable iterated multiveblen coniigurations may start from [G.sup.*.sub.2](4).

Proposition A.3. Let [X.sub.0] = {1,2,3,4}, [p.sup.1] = {5,6}, [X.sub.1] = [X.sub.0] [union] [p.sup.1], and [p.sup.2] = {7,8}.

Assume that [P.sub.0] [approximately equal to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The structure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] cannot be embedded into a Desarguesian projective space for any graph [P.sub.1] defined on [X.sub.1].

Proof. Write M = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that M is embedded into a Desarguesian projective space B, then from A.2 B satisfies the Fano axiom, because M contains [R.sup.*.sub.4] embedded into B. Observe the quadrangle [c.sub.{15}], [c.sub.{16}], [c.sub.{2,5}], [c.sub.{2,6}]. Its diagonal points in B are the following: [p.sup.1] = [c.sub.{5,6}], [c.sub.{1,2}], and [c.sup.*.sub.{1,2}] = [c.sub.{3,4}]; thus L([p.sup.1], [c.sub.{1,2}], [c.sub.{3,4}]). Analogously we obtain L([p.sup.1], [c.sub.{1,3}], [c.sub.{2,4}]), and therefore the point [p.sup.1] is the common point of the lines [bar.[C.sub.{1,2}],[c.sub.{3,4}] and [bar.[c.sub.{13}], [c.sub.{2,4}]]. Next, let us observe the substructure N of M spanned by the lines that pass through [p.sup.2] and have numbers in [X.sub.0]; it is embedded into the same projective space B and it is seen that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This yields that (in particular) [P.sub.1] [??] [X.sub.0] [approximately equal to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; what is more important, analogous reasoning applied to the quadrangles [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2] and [a.sub.1], [a.sub.3], [b.sub.1], [b.sub.3] gives that [p.sup.2] is the common point of the lines [bar.[C.sub.{12}]/c.sub.{3,4}]] and [bar.[c.sub.{1,3}]/[c.sub.{2,4}]. This finally gives [p.sup.1] = [p.sup.2], which is impossible.

References

[1] A. Beutelspacher, L. M. Batten, The theory of Unite linear spaces, University Press, Cambridge 1993.

[2] H. S. M. Coxeter, Desargues configurations and their collineation groups, Math. Proc. Camb. Phil. Soc. 78(1975), 227-246.

[3] I. Golonko, M. Prazmowska, K. Prazmowski, Adjacency in generalized projective Veronese spaces, Abh. Math. Univ. Hamburg 76 (2006), 99-114.

[4] H. Gropp, Configurations and their realization, Combinatorics (Rome and Montesilvano, 1994), Discrete Math. 174, no. 1-3 (1997), 137-151.

[5] D. Hilbert, P. Cohn-Vossen, Anschauliche Geometrie, Springer Verlag, Berlin, 1932.

[6] S. Kantor, Die Konfigurationen (3,3)10, Sitzungsberichte Wiener Akad. 84 (1881), 1291-1314.

[7] H. van Maldeghem, Slim and bislim geometries, in Topics in diagram geometry, 227-254, Quad. Mat., 12, Dept. Math., Seconda Univ. Napoli, Caserta 2003.

[8] A. Naumowicz, K. Prazmowski, The geometry of generalized Veronese spaces, Result. Math. 45 (2004), 115-136.

[9] M. Prazmowska, K. Prazmowski, Combinatorial Veronese structures, their geometry, and problems ofembeddability, Result. Math., 51(2008), 275-308.

[10] M. Prazmowska, Multiple perspective and generalizations of Desargues configuration, Demonstratio Math. 39(2006), no. 4,887-906.

[11] M. Prazmowska, K. Prazmowski, A generalization of Desargues and Veronese configurations, Serdica Math. J. 32(2006), no. 2-3,185-208.

[12] M. Prazmowska, Twisted projective spaces and linear completions of some partial Steiner triple systems, Algebra Geom. Comm. 49(2008), no 2, 341-368.

[13] M. Scafati, G. Tallini, Semilinear spaces and their remarkable subsets, J. Geom. 56 (1996), 161-167.

[14] E. Steinitz, Konfigurationen der projektiven Geometrie, Encyclop. d. math. Wiss. III (Geometrie) 1910,481-516.

[15] G. Tallini, Partial linear spaces and algebraic varieties, Symp. Math., 28 (1986), 203-217.

[16] R. W. Wilson, Introduction to graph theory, Addison Wesley Longman Limited, 1996.

Institute of Mathematics

University of Biatystok

ul. Akademicka 2

15-267 Biatystok, Poland

e-mail: malgpraz@math.uwb.edu.pl

Received by the editors February 2007 - In revised form in April 2009. Communicated by J. Thas. 2000 Mathematics Subject Classification : 51A45, (51E10,51E20).

(1) Formally, it consists of four pairwise intersecting lines, no three concurrent, together with the corresponding intersection points.

(2) See e.g. [12] for a detailed discussion of the case when the degree n of p is 4 - there are at least 11 distinct isomorphism types of such structures.

(3) Classical Grassmann space, as considered in geometry, is an incidence structure defined over the lattice of subspaces of a projective space, cf. [15]. Its points are the k-dimensional subspaces (k [greater than or equal to] 1 is a fixed integer) and its lines are the pencils. If k +1 is less than the dimension of the space then an equivalent structure is obtained when we adopt the (k + 1)-dimensional subspaces as the lines (and inclusion as the incidence). Passing to the lattice of subsets of a set X and replacing dimension by cardinality we define, by analogy, the combinatorial Grassmannian [G.sub.k](X).

(4) This terminology may be justified by the fact that Vo is isomorphic with one of the combinatorial Veronese spaces, cf. [9], [8] (which on the other hand, generalize classical projective Veronese spaces, cf. [15], [3]). In general, (see [11, Prop. 6]) the incidence structure B(n) is isomorphic to the dual of a suitable combinatorial Veronese space (i.e. isomorphic to [V.sup.*.sub.n](3) dual to [V.sub.n](3) in the notation of [9]).

(5) Note, that in the case of P = NX Proposition 2.3 is also a direct consequence of [9, Theorem 5.10].

(6) From a more general perspective, the existence of the required embedding is a consequence of some results on linear completions of multiveblen coniigurations: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and = [G.sub.2](4) = [G.sub.2](6) have the common linear completion: the projective Fano 3-space, cf. [12].
COPYRIGHT 2010 Belgian Mathematical Society
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2010 Gale, Cengage Learning. All rights reserved.

 
Article Details
Printer friendly Cite/link Email Feedback
Author:Prazmowska, Malgorzata
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Article Type:Report
Geographic Code:4EXPO
Date:May 1, 2010
Words:8461
Previous Article:On subordinations for certain analytic functions associated with Fox-Wright psi function.
Next Article:Application of duality techniques to starlikeness of weighted integral transforms.
Topics:

Terms of use | Privacy policy | Copyright © 2018 Farlex, Inc. | Feedback | For webmasters