SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR

Algebra i analiz Tom 21 (2009),
1

St. Petersburg Math. J. Vol. 21 (2010), No. 1, Pages 65–93 S 1061-0022(09)01086-3 Article electronically published on November 4, 2009

SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS S. EVDOKIMOV AND I. PONOMARENKO Abstract. There are several schemes (coherent configurations) associated with a finite projective plane P. In the paper, a new scheme is constructed, which, in a sense, contains all of them. It turns out that this scheme coincides with the 2extension of the nonhomogeneous scheme of P and is uniquely determined up to similarity by the order q of P. Moreover, for q ≥ 3, the rank of the scheme does not depend on q and equals 416. The results obtained have interesting applications in the theory of multidimensional extensions of schemes and similarities.

§1. Introduction A projective plane is a triple consisting of a set of points, a set of lines, and an incidence relation between their elements, the defining property of which is the following: any two distinct points (respectively, lines) are incident to a unique line (respectively, point). To avoid degenerate cases, usually a certain nondegeneracy condition is also imposed. An isomorphism of projective planes is a bijection taking the points (respectively, lines) of one plane to the points (respectively, lines) of the other and preserving the incidence relation. In this paper we deal with finite projective planes. For such a plane, there exists a positive integer q, the order of the plane, such that each point (respectively, line) is incident to exactly q + 1 lines (respectively, points), and the number of points equals the number of lines and equals q 2 + q + 1 (see [14]). For any prime power q > 1, the projective plane whose points and lines are those of a 3-dimensional linear space over a Galois field of order q with the incidence relation defined by inclusion, is called a Galois plane of order q.1 The Galois planes are precisely the finite projective planes for which the Desargues theorem is true. Though there are a lot of non-Desarguesian finite projective planes, up to now it is not known if there exists such a plane of prime order. There are several schemes (coherent configurations) associated with a finite projective plane P: a scheme of type [ 2 22 ] (nonhomogeneous scheme), a scheme of a distanceregular graph of diameter 3 (homogeneous scheme), and schemes of flags and antiflags [13, 4, 12, 8]. In this paper we construct a new scheme that, in a sense, contains all of them. The underlying set of it equals V 2 , where V is the set of elements (the points and lines) of the plane P; in the Desarguesian case it is a fusion of the scheme corresponding to the coordinatewise action of the group Aut(P) on V 2 . Moreover, it turns out that in the sense of [6] our scheme is none other than the 2-extension of the nonhomogeneous 2000 Mathematics Subject Classification. Primary 05C25, 51A05. Key words and phrases. Projective plane, Galois plane, scheme, graph. The first author was partially supported by RFBR (grants 07-01-00485 and 06-01-00471). The second author was partially supported by RFBR (grants 07-01-00485 and 05-01-00899) and by the grant NS-4329.2006.1. 1 Here the “plane of order 1” (triangle) is also treated as a Galois plane. c
2009 American Mathematical Society

65

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S. EVDOKIMOV AND I. PONOMARENKO

scheme of P. This fact has interesting applications in the theory of multidimensional extensions of schemes and similarities. The basis relations of the new scheme mentioned above will be constructed with the help of a special complete colored graph Γ with a vertex set Ω such that Ξ ⊂ Ω ⊂ Ξ ∪ Ξ(2) , where Ξ is the set consisting of four distinct commuting variables ξ1 , ξ2 , ξ3 , ξ4 , and Ξ(2) is the symmetric square of Ξ, the elements of which are written as products of two distinct variables. The colors are called types and come from the types of elements and relations among them in a projective plane. Each vertex of Γ has type p (point) or l (line), whereas each edge has type e (equality), e
(inequality), i (incidence) or i
(nonincidence). We also assume some natural compatibility conditions: (a) the edge between vertices of coinciding (respectively, distinct) types must have type e or e
(respectively, i or i
), (b) if an edge between variables ξi and ξj is of type e
, then ξi ξj ∈ Ω and the edge between ξi and ξi ξj is of type i. Any such graph Γ is called a Ξ-configuration. Let x = (x1 , . . . , x4 ), where xi ∈ V for all i. With x we associate a Ξ-configuration as follows. Let Ω be the union of Ξ and the set of all ξi ξj ∈ Ξ(2) such that xi and xj are distinct points or distinct lines in P. The type of a vertex ξ ∈ Ω (respectively, an edge between ξ and η) is defined to be the type of the element f (ξ) (respectively, the relation between f (ξ) and f (η)) in the plane P, where f (ξ) = xi for ξ = ξi and f (ξ) equals the (unique) element of P incident to both xi and xj for ξ = ξi ξj . The graph Γ constructed in this way is denoted by Der(x) = Der(x, P). For a Ξ-configuration Γ, set R(Γ, P) = {x ∈ V 4 : Der(x) = Γ}. Clearly, when Γ runs over all Ξ-configurations, the nonempty sets of this kind form a partition Π of V 4 called the derivative partition associated with P. Obviously, for a given set R ∈ Π, the graph Der(x) does not depend on the choice of x ∈ R. We denote this graph by Der(R) and set Der(P) = {Der(R) : R ∈ Π}. Below, classes of the partition P are identified with binary relations on V 2 via the natural bijection from V 4 to V 2 × V 2 that takes (x1 , x2 , x3 , x4 ) to ((x1 , x2 ), (x3 , x4 )). Theorem 1.1. Let P be a finite projective plane of order q and Π the derivative partition associated with P. Then: 1) the pair (V 2 , Π) is a scheme; 2) the set Der(P) does not depend on P for a fixed q, and even does not depend on q for sufficiently large q; 3) for given R, S, T ∈ Π, the intersection number cTR,S of the scheme (V 2 , Π) equals f (q), where f (x) is a polynomial with integral coefficients depending only on Der(R), Der(S), and Der(T ) (but not depending on P). The Ξ-configurations belonging to the set Der(P) are described in the table of the Appendix. It follows that this set does not depend on q for q ≥ 3, and the cardinality of Π equals 200, 400, and 416 for q = 1, q = 2, and q ≥ 3, respectively. The reflexive relations of Π are given by the eight Ξ-configurations uniquely determined by their restrictions to the set {ξ1 , ξ2 }: p

l

e

e

p

p

l

l

e


e


p

p

l

l

i

i

l

p

p

l

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i


i


l

p

SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

67

Thus, the scheme (V 2 , Π) has eight homogeneity sets: (1)

Et = {(x, y) ∈ Vt × Vt : x is equal to y},

Et
= (Vt × Vt ) \ Et ,

(2)

It = {(x, y) ∈ Vt × Vt
: x is incident to y},

It
= (Vt × Vt
) \ It ,

where t ∈ {p, l}, t
is determined by the condition {t, t
} = {p, l}, and Vp and Vl are the points and the lines of P. These eight sets, viewed as binary relations on V , form a partition R of V 2 , and the pair (V, R) is exactly the scheme introduced by D. Higman in [13]. We call (V, R) and (V 2 , Π) the (nonhomogeneous) scheme and the derivative scheme of the projective plane P, respectively. The mapping x → (x, x) takes the basis relations of the former scheme to the relations R(Γ, P), where Γ runs over the Ξconfigurations obtained from the above eight Ξ-configurations by interchanging ξ2 and ξ3 . Thus, the nonhomogeneous scheme is isomorphic to the restriction of the derivative scheme to Ep ∪ El . Similarly, one can see that the restrictions of the derivative scheme to Ip and Ip
are the scheme of flags (see [12]) and an extension of the scheme of antiflags, respectively (see [8]). From Theorem 1.1 it follows that for a sufficiently large q equal to the order of a finite projective plane P, there exists a uniquely determined noncommutative algebra over R of dimension 416, the adjacency algebra of the derivative scheme of P. The elements of the standard linear basis of it are parametrized by the Ξ-configurations from the set Der(P), not depending on q by statement 2) of the theorem; the structure constants are given by a tensor T (q), the entries of which are determined by statement 3). Since any prime power is the order of some projective plane, for every real q > 0 there exists an algebra with structure constant tensor T (q). Moreover, this algebra can be regarded as a generalized C-algebra in the sense of [1]. Theorem 1.1 also has interesting applications in the theory of m-extensions of schemes and similarities, as developed in [7, 5, 6, 2]. By definition, the m-extension of a scheme C on V is a special scheme on the set V m , whereas the m-extension of a similarity between schemes C and C
is a special similarity between their m-extensions (see §3 for the details). It is known that (3) C = Cs(1) ≤ · · · ≤ Cs(n) = · · · = Cs(∞) , where Cs(m) is the m-closure of C (i.e., the restriction of the m-extension to the diagonal of V m identified with V ), Cs(∞) is the scheme of the group Aut(C), and n = |V |, and that (4)

Sim(C, C
) = Sim1 (C, C
) ⊃ · · · ⊃ Simn (C, C
) = · · · = Sim∞ (C, C
),

where Simm (C, C
) is the set of all m-similarities from C to C
(i.e., the similarities admitting an m-extension) and Sim∞ (C, C
) is the set of all similarities from C to C
induced by isomorphisms of these schemes. In fact, the starting point of this paper was to determine the 2-extension of the scheme of a projective plane and to find a sufficient condition for the latter scheme to be 2-closed. Generally, the problem of finding the m-extension of a scheme seems to be very hard. Apart from trivial cases, it was completely solved only for cyclotomic schemes over a finite field [2]. In our case the answer is given by the following statement. Theorem 1.2. Let C be a scheme of a finite projective plane P, and let Cp be its 2extension. Then: 1) Cp equals the derivative scheme of P; 2) the scheme C is 2-closed. Let C and C
be schemes of finite projective planes P and P
of the same order with the sets of basis relations R and R
, respectively. Then the bijection from R to R
preserving the types of basis relations gives a similarity ϕ : C → C
called the canonical one. By

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68

S. EVDOKIMOV AND I. PONOMARENKO

statement 2) of Theorem 1.1, we have Der(P) = Der(P
). So the mapping ϕ p from Π to Π
such that R(Γ, P)ϕp = R(Γ, P
), Γ ∈ Der(P), where Π and Π
are the derivative partitions associated with P and P
, respectively, is a bijection. Thus, from statement 3) of Theorem 1.1 and statement 1) of Theorem 1.2 it follows that ϕ p is a similarity from Cp to Cp
, which is obviously the 2-extension of ϕ. This proves the following result. Theorem 1.3. The canonical similarity between the schemes of two finite projective planes of the same order is in fact a 2-similarity. There are natural analogs of Theorems 1.1 and 1.3 for the homogeneous scheme of a projective plane P defined as follows. Let C be the nonhomogeneous scheme of the plane P. Then it is easily seen that the group Φ = Sim(C, C) consists of two elements. Denote by ϕ the nontrivial one. Clearly, ϕ coincides with the canonical similarity from C to the scheme of the projective plane dual to P. (So, by Theorem 1.3, the similarity ϕ has a 2-extension ϕ.) p The scheme D = C Φ obtained from C by merging the relations in each Φ-orbit is called the homogeneous scheme of P. One can see that D is the scheme of the incidence graph of P, which is a bipartite distance-regular graph of diameter 3. If C
is the scheme of a projective plane P
of the same order as P, then the canonical similarity from C to C
induces by restriction a similarity ψ : D → D
, where D
is the homogeneous scheme of P
. Being, obviously, the only element of Sim(D, D
), the similarity ψ is called the canonical similarity from D to D
. Theorem 1.4. In the above notation we have: p = {1 p, ϕ}; p = CpΦp , where Φ 1) D C p 2) the scheme D is 2-closed ; 3) the canonical similarity from D to the homogeneous scheme of another projective plane of the same order is in fact a 2-similarity. In the study of permutation groups of rank 3, for strongly regular graphs Higman [11] introduced the notion of a t-condition, which was generalized to arbitrary colored graphs in [9] (see also [6]). Below, by the colored graph of a scheme (V, R) we mean a graph, the vertex set and the set of colored classes of which are V and R, respectively. By [6, Theorem 6.4], the colored graph of any 2-closed scheme satisfies the 6-condition. So, statement 2) of Theorem 1.4 implies the following result, which shows in particular that there are infinitely many distance-regular graphs of diameter 3 that satisfy the 6-condition and are not distance-transitive. Corollary 1.5. The colored graph of the scheme (homogeneous or not) of any finite projective plane satisfies the 6-condition. In the framework of the theory of m-extensions, the most important invariants of a scheme C are the Schurity number t(C) and the separability number s(C) introduced in [6] with the help of inclusions (3) and (4), respectively (see §3). Let us analyze these invariants in the case where C is the (homogeneous or not) scheme of a projective plane P of order q. We observe that, by the Ostrom–Wagner theorem [14, Theorem 14.13], the plane P is a Galois plane if and only if its group of collineations acts 2-transitively on the points of P. Thus, t(C) = 1 if and only if P is a Galois plane (cf. [6, Theorem 7.9]). Moreover, it is easily seen that s(C) = 1 if and only if any projective plane of order q is isomorphic to P in the nonhomogeneous case, and either to P or to its dual plane in the homogeneous case. Next, the definitions show that if t(C) = 2 (respectively, s(C) = 2), then C < Cs(∞) = s (2) C (respectively, Sim(C, C
)
Sim∞ (C, C
) = Sim2 (C, C
) for some scheme C
). However,

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

69

if C is the nonhomogeneous scheme of a projective plane, this contradicts statement 2) of Theorem 1.2 (respectively, statement 2) of Theorem 1.3), whereas if C is homogeneous, this contradicts statement 2) of Theorem 1.4 (respectively, statement 3) of Theorem 1.4). Thus, we arrive at the following result. Corollary 1.6. The Schurity and separability numbers of the scheme (homogeneous or not) of a projective plane cannot equal 2. In [6, Theorem 7.9] it was proved that s(C) ≤ 6 whenever C is a homogeneous scheme of a Galois plane of order q. The following theorem generalizes and refines this result. From Corollary 1.6 it follows that the new upper bound is attained whenever there exists a non-Galois plane of order q (there are infinitely many such numbers q; see [14]). Theorem 1.7. Let C be the scheme (homogeneous or not) of a Galois plane. Then s(C) ≤ 3. The proof of Theorem 1.1 occupies §§4–8. In the first two of those sections, we introduce the concept of a configuration (generalizing that of a Ξ-configuration) and study general properties of configurations as well as their embeddings in projective planes.2 The proof itself is in §6. The key point is to show that the numbers of embeddings of special configurations in a projective plane of order q depend only on q; these numbers are in fact the intersection numbers of the derivative scheme. The configurations in question turn out to be admissible in the sense of §7 (Theorem 7.3). In §8 we prove that the embedding number of any admissible configuration can be expressed via those of admissible configurations of smaller size (Theorem 7.4). The proofs of Theorems 1.2, 1.4, and 1.7 are contained in §§9, 10, and 11, respectively. In the Appendix, for any projective plane we present the list of Ξ-configurations parameterizing the basis relations of the derivative scheme. To make the paper self-contained as far as possible, we cite the background on schemes and their multidimensional extensions in §§2 and 3 (the details can be found in [1]). Part of the results related to homogeneous schemes of projective planes was announced in [3]. Concerning the theory of finite projective planes we refer the reader to [14]. Notation. As usual, we denote by Z and R the ring of integers and the field of reals, respectively. The cardinality of a finite set V is denoted by |V |, and the diagonal of V 2 by ∆(V ). For a (binary) relation R on V and v ∈ V , we set RT = {(u, v) ∈ V 2 : (v, u) ∈ R}, Rin (v) = {u ∈ V : (u, v) ∈ R},

Rout (v) = {u ∈ V : (v, u) ∈ R}.

For relations R, S ⊂ V , we set 2

R · S = {(u, w) ∈ V 2 : (u, v) ∈ R, (v, w) ∈ S for some v ∈ V }, R ⊗ S = {((u1 , u2 ), (v1 , v2 )) ∈ (V 2 )2 : (u1 , v1 ) ∈ R, (u2 , v2 ) ∈ S}. For an equivalence relation E on V , we denote by V /E the set of its classes. If R is a relation on V , then we set RV /E = {(X, Y ) ∈ (V /E)2 : RX,Y = ∅}, where RX,Y = R ∩ (X × Y ). For X ⊂ V the set RX = RX,X is treated as a relation on X. The group of all permutations of V is denoted by Sym(V ). 2 The

idea to use configurations was inspired by the paper [10].

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70

S. EVDOKIMOV AND I. PONOMARENKO

Each bijection f : V → V
(v → v f ) determines naturally a bijection R → Rf from the relations on V onto the relations on V
and a group isomorphism g → g f from Sym(V ) onto Sym(V
). For an equivalence relation E on V , the bijection f induces a bijection fV /E : V /E → V
/E
, where E
= E f . For i = 1, . . . , m, the ith coordinate of x ∈ V m is denoted by xi . The set of elements (points and lines) of a projective plane P is denoted by V . To define the type of an element of P, we introduce the two-element set T1 = {p, l} and a function t1,P : V → T1 . By definition, t1,P (x) = p (respectively, t1,P (x) = l) if and only if x is a point (respectively, line) of the plane P. The elements p and l are said to be dual to each other. To define the type of a relation between two elements of P, we introduce the fourelement set T2 = {e, e
, i, i
} and a function t2,P : V 2 → T2 . By definition t2,P (x, y) = e (respectively, t2,P (x, y) = e
) if and only if x, y are equal (respectively, unequal) elements of the same type; similarly, t2,P (x, y) = i (respectively, t2,P (x, y) = i
) if and only if x, y are incident (respectively, nonincident) elements of distinct types. When it does not lead to misunderstanding, we omit the subscript P in t1,P and t2,P . For distinct elements x, y of P of the same type, the (only) element incident to both of them is denoted by xy. §2. Schemes and similarities Let V be a finite set and R a partition of V 2 closed with respect to transposition. Denote by R∗ the set of all unions of elements of R. A pair C = (V, R) is called a coherent configuration or a scheme on V if the diagonal ∆(V ) of V 2 belongs to R∗ and if, for any R, S, T ∈ R, the number

(5) cR,S (u, w) =
{v ∈ V : (u, v) ∈ R, (v, w) ∈ S}
does not depend on the choice of (u, w) ∈ T . The elements of the sets V , R = R(C), R∗ = R∗ (C), and the numbers (5) are called the points, the basis relations, the relations and the intersection numbers of C, respectively; the intersection numbers are denoted by cTR,S . From the definition it follows that R · S ∈ R∗ for all R, S ∈ R∗ . The number rk(C) = |R| is called the rank of C. The set V is the disjoint union of homogeneity sets of C, i.e., of sets X ⊂ V for which ∆(X) ∈ R. For any R ∈ R, two homogeneity sets X and Y such that R ⊂ X × Y exist ∆(X) ∆(Y ) and are uniquely determined. Put dout (R) = cR,RT and din (R) = cRT ,R . Then dout (R) = |Rout (x)|,

din (R) = |Rin (y)|,

x ∈ X, y ∈ Y.

By a set of C we mean any union of homogeneity sets. It is easily seen that X and Y are sets of C if and only if X × Y ∈ R∗ . The scheme C is called homogeneous if V is a homogeneity set of it. We say that two schemes are isomorphic if there exists a bijection between their point sets preserving the basis relations. Any such bijection is called an isomorphism of these schemes. The group of all isomorphisms of a scheme C contains a normal subgroup Aut(C) = {f ∈ Sym(V ) : Rf = R, R ∈ R} called the automorphism group of C. On the other hand, let G ≤ Sym(V ) be a permutation group and R the set of orbits of the coordinatewise action of G on V 2 . Then (V, R) is a scheme; we call it the scheme of G. Given a set X ⊂ V , we denote by RX the set of all nonempty relations RX , R ∈ R. If X is a set of C, then CX = (X, RX ) is a scheme. Given an equivalence relation E on V , we denote by RV /E the set of all nonempty relations RV /E , R ∈ R. If E ∈ R∗ , then

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

71

CV /E = (V /E, RV /E ) is a scheme. The set of all equivalence relations E ∈ R∗ is denoted by E. For schemes C1 = (V1 , R1 ) and C2 = (V2 , R2 ), let R1 ⊗R2 denote the set of all relations R1 ⊗ R2 , R1 ∈ R1 , R2 ∈ R2 . Then the pair C1 ⊗ C2 = (V1 × V2 , R1 ⊗ R2 ) is a scheme. It is called the tensor product of C1 and C2 . The set of all schemes on V is partially ordered by inclusion, namely, C ≤ C
if and only if R∗ ⊂ (R
)∗ ; in this case, we call C
an extension of C. For sets R1 , . . . , Rs of binary relations on V , we denote by [R1 , . . . , Rs ] the smallest scheme C on V such that Ri ⊂ R∗ for all i; we omit the square brackets if Ri = {Ri } and write Ci instead of Ri if the latter is the set of basis relations of Ci . For an equivalence relation E on V we set CE = [C, {∆(X) : X ∈ V /E}]. Schemes C and C
are said to be similar if ϕ

cTR,S = cTRϕ ,S ϕ ,

(6)




R, S, T ∈ R,

for some bijection ϕ : R → R , R → R , called a similarity from C to C
. The set of all similarities is denoted by Sim(C, C
). In a natural way, each isomorphism from C to C
induces a similarity between these schemes. The set of all isomorphisms from C to C
inducing a similarity ϕ is denoted by Iso(C, C
, ϕ). A similarity ϕ induces bijections both between the sets and between the relations of C and C
; we use the same letter ϕ to denote these bijections. It can be seen that E ϕ = E
, V ϕ = V
, and (7)

(RT )ϕ = (Rϕ )T ,

ϕ

dout (R) = dout (Rϕ ),

din (R) = din (Rϕ ),

R ∈ R.

For any E ∈ E, the similarity ϕ induces a similarity ϕV /E : CV /E → CV

/E
, where E
= E ϕ . Finally, for a group Φ ≤ Sim(C, C), we set 

RΦ = {RΦ : R ∈ R},

where RΦ = ϕ∈Φ Rϕ . Then the pair C Φ = (V, RΦ ) is a scheme. A scheme C = (V, R) is semiregular if dout (R) ≤ 1 and din (R) ≤ 1 for all R ∈ R. For homogeneous schemes, the first three statements of the following theorem were proved in [15]. Theorem 2.1. Let C = (V, R) be a scheme, and let E ∈ E. Suppose that the scheme CV /E is semiregular. Then: 1) R(CE ) = {RX,Y : R ∈ R, X, Y ∈ V /E, RX,Y = ∅}; 2) for every g ∈ Aut(CV /E ), the mapping ψg : RX,Y → RX g ,Y g is a similarity of CE ; 3) C = (CE )Ψ , where Ψ = {ψg : g ∈ Aut(CV /E )};
4) for any similarity ϕ : C → C
there exists a similarity ϕE : CE → CE
extending
ϕ
ϕ, where E = E ; moreover, for each f ∈ Iso(CV /E , CV
/E
, ϕV /E ) the similarity ϕE can be chosen so that (ϕE )V /E is induced by f . Proof. First, we prove that, given R, S, T ∈ R and X, X
, X

, Y, Y
, Y

∈ V /E with RX,Y = ∅ and SX
,Y
= ∅, we have (8)

T cRX ,Y ,SX
,Y
(u, v) = δX

,X δY,X
δY
,Y

cR,S ,

(u, v) ∈ TX

,Y

.

Without loss of generality we assume that X

= X, Y = X
, and Y
= Y

. Since RX ,Y = ∅ and SX
,Y
= ∅, the semiregularity of CV /E implies that (RV /E )out (X) = {Y } and (SV /E )in (Y
) = {X
} for some X
, Y ∈ V /E. So, cR,S (u, v) = cRX ,Y ,SX
,Y
(u, v),

(u, v) ∈ X × Y
.

If (u, v) ∈ T , then the left-hand side equals cTR,S , and (8) follows.

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72

S. EVDOKIMOV AND I. PONOMARENKO

Denote by R
the right-hand side in statement 1) of the theorem. Clearly, R
is a partition of V 2 closed with respect to transposition, and ∆(V ) ∈ (R
)∗ . Therefore, (8) implies that C
= (V, R
) is a scheme, the intersection numbers of which look like this: T









,Y T cRXX,Y ,S = δX

,X δY,X
δY
,Y

cR,S . X
,Y


(9)

Since, obviously, C
≥ C and CE ≥ C
, we conclude that C
= CE . Statement 1) is proved. Statement 2) follows immediately from (9) and the definition of ψg . Finally, by [6, Theorem 4.4], the automorphism group of a semiregular scheme acts transitively on any of its basis relations. Thus, given R ∈ R, we have   RX,Y = RX0g ,Y0g , R= (X,Y )∈RV /E

g∈Aut(CV /E )

where X0 , Y0 ∈ V /E with RX0 ,Y0 = ∅. This proves statement 3). To prove statement 4), we let ϕ : C → C
be a similarity. Then CV

/E
coincides with the ϕV /E -image of CV /E and hence it is semiregular (see (7)). Therefore, Iso(CV /E , CV

/E
, ϕV /E ) = ∅ by [6, Theorem 4.4]. Take f ∈ Iso(CV /E , CV

/E
, ϕV /E ). By

(9), the mapping RX,Y → RX f ,Y f , RX,Y ∈ R(CE ) induces a similarity from CE to CE
. Since the restriction of it to C coincides with ϕ, we are done.
−1 Let P be a projective plane of order q, Vp = t−1 1,P (p) and Vl = t1,P (l) the sets of its points and lines, V = Vp ∪ Vl , and R the set of the following eight relations (cf. (1) and (2)):

Et = t−1 2,P (e)Vt ,Vt ,


Et
= t−1 2,P (e )Vt ,Vt ,

It = t−1 2,P (i)Vt ,Vt
,


It
= t−1 2,P (i )Vt ,Vt
,

t ∈ T1 ,

where t
is the dual to t (see the notation). Then it is easily seen that the pair C = (V, R) is a scheme on 2(q 2 + q + 1) points; it is called the (nonhomogeneous) scheme of P. This scheme has two homogeneity sets Vp and Vl and satisfies the following conditions with X = Vp , Y = Vl and R = Ip , or X = Vl , Y = Vp and R = Il : (P1) rk(C) = 8 and rk(CX ) = rk(CY ) = 2, (P2) cSR,RT = 1 for some basis relation R ⊂ X × Y and S = X 2 \ ∆(X). In fact, any scheme C with two homogeneity sets X and Y satisfying (P1) and (P2) is the scheme of some projective plane. We note that the degrees (din and dout ) of the relations Et , Et
, It , and It
equal 1, q 2 + q, q + 1, and q 2 (respectively), where t ∈ {p, l}. Let ϕ : C → C
be a similarity. From what has been said above and the properties of similarities, we show that C
is also the scheme of a projective plane of order q. Moreover, ϕ takes the basis relations of C either to the corresponding basis relations of C
, or to the dual ones.3 Conversely, any of these two mappings induces a similarity from C to C
(the first of them is said to be canonical). In particular, Φ = Sim(C, C) is a group of order 2. The scheme D = C Φ is called the homogeneous scheme of P. As above, any scheme similar to D is the homogeneous scheme of a projective plane of the same order. Moreover, for any two homogeneous schemes of projective planes of the same order, there exists a unique similarity between them called canonical. §3. Extensions of schemes and similarities Let C be a scheme on V and m a positive integer. By definition, the m-extension of C is the following scheme on V m : (10) 3 The

Cp(m) = [C m , ∆m ], relations Ep and El , Ep and El , Ip and Il , Ip and Il are said to be dual to each other.

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where C m is the m-fold tensor product of C and ∆m is the diagonal of V m . We observe that, except for trivial cases, the m-extension is a nonhomogeneous scheme for all m ≥ 2. The above definition implies that Cp(1) = C and Cp(m) is a fusion of the scheme corresponding to the coordinatewise action of the group Aut(C) on V m . Moreover, Aut(C) ∼ = Aut(Cp(m) )∆m ∼ = Aut(Cp(m) ). For any m the intersection numbers of the m-extension of C are invariants of it. Namely, (m) a similarity ψ : Cp(m) → Cp
is called an m-extension of a similarity ϕ : C → C
if 1) (∆m )ψ = ∆
m , m 2) Rψ = Rϕ for all R ∈ R(C m ), where ϕm : C m → (C
)m is the m-fold tensor product of ϕ. This definition shows that each similarity has a 1-extension coinciding with it. Moreover, for any m, the existence of an m-extension of ϕ implies its uniqueness; we denote it by ϕ p(m) . Observe that not every similarity admits an m-extension. However, if ϕ is induced by some isomorphism, then ϕ has an m-extension for all m. Set (11)

−1 Cs = Cs(m) = ((Cp(m) )∆ )δ ,

where δ : v → (v, . . . , v) is the diagonal mapping from V to V m . The scheme Cs is called the m-closure of C; if C = Cs(m) , then we say that C is m-closed. A similarity between two schemes is called an m-similarity if it admits an m-extension. Any such similarity ϕ is extended uniquely to a similarity ϕ s(m) between the m-closures. The set of all m-similarities from C to C
is denoted by Simm (C, C
). The following theorem was proved in [5]. Theorem 3.1. For an arbitrary scheme C on n points, we have (12) (13)

C = Cs(1) ≤ · · · ≤ Cs(n) = · · · = Cs(∞) , Sim(C, C
) = Sim1 (C, C
) ⊃ · · · ⊃ Simn (C, C
) = · · · = Sim∞ (C, C
),

where Cs(∞) is the scheme of the group Aut(C) and Sim∞ (C, C
) is the set of all similarities from C to C
induced by isomorphisms of these schemes. A scheme C is said to be m-Schurian if Cs(m) = Cs(∞) and m-separable if Simm (C, C
) = Sim∞ (C, C
) for all C
. The integers t(C) = min{m : C is m-Schurian},

s(C) = min{m : C is m-separable}

are called the Schurity number and the separability number of C, respectively. It can be checked that t(C) = s(C) = 1 whenever the scheme C is semiregular; see [6, Theorem 4.4]. The following theorem will be used throughout the paper; it is a straightforward consequence of [6, Lemma 6.2]. Below, given R ⊂ V 2 and i, j ∈ {1, . . . , 2m}, we set CylR (i, j) = {x ∈ V 2m : (xi , xj ) ∈ R}. Theorem 3.2. Let C be a scheme on V , and let Cp = Cp(m) and Cs = Cs(m) , where m is a positive integer. Then for any relation R of the scheme Cs and any i, j ∈ {1, . . . , 2m}, the following statements are true: p 1) the set CylR (i, j) is a relation of the scheme C; 2) CylR (i, j)ϕp = CylRϕs (i, j) for any m-similarity ϕ : C → C
, where ϕ p=ϕ p(m) and ϕ s=ϕ s(m) .

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74

S. EVDOKIMOV AND I. PONOMARENKO

§4. Configurations Let Ω be a finite set. We shall consider colored undirected graphs on Ω with loops; the vertices and the edges of these graphs will be labeled with elements of the sets T1 and T2 , respectively (see the notation). The edge set is identified with a symmetric binary relation E on Ω such that ∆(Ω) ⊂ E ⊂ Ω2 . The coloring is given by the functions t1 : Ω → T1 ,

t2 : E → T2 .

Thus, such a graph Γ is represented by the triple (Ω, E, t), where t = (t1 , t2 ). Definition 4.1. A colored graph Γ = (Ω, E, t) is called a configuration on Ω if ∆(Ω) ⊂ Ee and Ee ∪ Ee
⊂ (Ωp × Ωp ) ∪ (Ωl × Ωl ),

Ei ∪ Ei
⊂ (Ωp × Ωl ) ∪ (Ωl × Ωp ),

−1 where Ωt = t−1 1 (t) for t ∈ T1 and Et = t2 (t) for t ∈ T2 .

We say that a configuration Γ is complete if E = Ω2 . Given a set Ω
⊂ Ω, the configuration obtained by restriction of Γ to Ω
is denoted by ΓΩ
. Let P be a finite projective plane. Then there is a natural complete configuration on the set V of its elements, where t1 = t1,P and t2 = t2,P (see the notation). Let i0 : Ω0 → V be a mapping, where Ω0 ⊂ Ω. Definition 4.2. A mapping i : Ω → V is called a strict Γ-extension of i0 if i|Ω0 = i0 and t1 (ξ) = t1,P (i(ξ)),

ξ ∈ Ω,

and

t2 (ξ, η) = t2,P (i(ξ), i(η)),

(ξ, η) ∈ E.

If the latter identity holds true only if t2 (ξ, η) ∈ {e, i}, then the mapping i is called a nonstrict Γ-extension. The set and the number of all strict Γ-extensions (respectively, nonstrict Γ-extensions) of a mapping i0 : Ω0 → V are denoted by Embstr (Γ, i0 , P) and estr (Γ, i0 , P) (respectively, by Emb(Γ, i0 , P) and e(Γ, i0 , P)). If Ω0 = ∅, we omit i0 and call the elements of Embstr (Γ, P) and Emb(Γ, P) strict and nonstrict embeddings of the configuration Γ into the plane P. Clearly, Embstr (Γ, i0 , P) ⊂ Emb(Γ, i0 , P) and Embstr (Γ, i0 , P) ⊂ Embstr (Γ, P),

Emb(Γ, i0 , P) ⊂ Emb(Γ, P)

for all i0 . Moreover, if Embstr (Γ, i0 , P) = ∅ (respectively, Emb(Γ, i0 , P) = ∅), then i0 is a strict (respectively, nonstrict) embedding of ΓΩ0 into P. We observe that the set Emb(Γ, i0 , P) does not change if we remove some edges of Γ belonging to the set E
= (Ee
∪ Ei
) \ Ω20 . Below, the elements e and e
, as well as i and i
, of the set T2 (see the notation) will be called opposite to each other. Theorem 4.3. In the above notation, if i0 ∈ Embstr (ΓΩ0 , P), then  estr (Γ, i0 , P) = (−1)|S| e(ΓS , i0 , P), S⊂E


where ΓS is obtained from Γ by replacing the type of any edge from S with its opposite. Proof. The definitions of strict and nonstrict extensions imply that  Emb(Γ, i0 , P) \ Embstr (Γ, i0 , P) = Emb(Γ{s} , i0 , P). 

s∈E


Since Γ∅ = Γ and Emb(ΓS , i0 , P) = s∈S Emb(Γ{s} , i0 , P) for all nonempty sets S ⊂ E
, the required statement follows by the inclusion-exclusion principle.


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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

75

Let Γ be a configuration on Ω. The connectivity components of the graph (Ω, Ee ) s denote the set of classes of this partition. Clearly, all induce a partition of Ω. Let Ω s elements of a class A ∈ Ω have the same type; we denote it by t1 (A). For a subset Ω0 of Ω, we set s0 = Ω s 0,Γ = {A ∈ Ω s : A ∩ Ω0 = ∅}. Ω s The class containing an element (vertex) ξ ∈ Ω is denoted by ξ. s are said to be incident if Two elements ξ, η ∈ Ω (respectively, classes A, B ∈ Ω) (ξ, η) ∈ Ei (respectively, (A × B) ∩ Ei = ∅). For ξ ∈ Ω and Ω0 ⊂ Ω (respectively, for s we let N (ξ, Ω0 ) (respectively, N (A, Ω s 0 )) be the set of all elements s and Ω s 0 ⊂ Ω), A∈Ω s of Ω0 (respectively, classes from Ω0 ) incident to ξ (respectively, to A); the cardinality s 0 )). If Ω0 = Ω (respectively, of this set is denoted by d(ξ, Ω0 ) (respectively, by d(A, Ω s s s Ω0 = Ω), we omit Ω0 (respectively, Ω0 ). Let Γ be a configuration on Ω, and let Ω0 ⊂ Ω, Γ0 = ΓΩ0 . Then the natural injection s 0,Γ → Ω s Γ that preserves the incidence of classes. from Ω0 to Ω induces a mapping f : Ω 0 We say that Ω0 is a Γ-subset of Ω if f is injective and two classes A0 , B0 of Γ0 are incident whenever the classes f (A0 ) and f (B0 ) of Γ are incident. The configuration Γ is said to sΓ = Ω s 0,Γ and the configuration Γ0 is complete. be final with respect to Ω0 if Ω Lemma 4.4. Let Γ = (Ω, E, t) be a final configuration with respect to a set Ω0 ⊂ Ω, and let i0 ∈ Embstr (Γ0 , P), where Γ0 = ΓΩ0 and P is a projective plane of order q. Then e(Γ, i0 , P) is equal to 0 or 1, and the latter possibility occurs if and only if Ω0 is a Γ-subset of Ω. Proof. Clearly, any nonstrict Γ-extension of i0 maps all elements of any class of Γ to one and the same element of P. Thus, the first statement follows from the finality of the configuration Γ. To prove the second statement, first we suppose that i is a nonstrict Γ-extension of i0 . Let A0 , B0 be classes of Γ0 , and let A = f (A0 ), B = f (B0 ). If A and B are equal (respectively, incident), then so are the elements i(A) = i0 (A0 ) and i(B) = i0 (B0 ) of P. Since the configuration Γ0 is complete, this implies that the classes A0 and B0 are equal (respectively, incident). Conversely, let Ω0 be a Γ-subset of Ω. Since s Γ . The injectivity of f shows that Γ is final, the set A0 = A∩Ω0 is nonempty for all A ∈ Ω A0 is a class of Γ0 . Define a mapping i : Ω → V , where V is the set of elements of P, by setting i(α) = i0 (A0 ), where A0 is as above with A being the class of Γ containing α. Clearly, i|Ω0 = i0 . Next, if (α, β) ∈ Ee (respectively, (α, β) ∈ Ei ), then the classes A and B are equal (respectively, incident), where A and B are the classes of Γ containing α and β, respectively. Since Ω0 is a Γ-subset of Ω, this implies that the classes A0 and B0 are equal (respectively, incident). Therefore, the elements i(α) = i0 (A0 ) and i(β) = i0 (B0 )
of P are equal (respectively, incident). Thus, i ∈ Emb(Γ, i0 , P). An important example of a configuration is given by any Ξ-configuration and, in particular, by the configuration Γ = Der(x, P), where x ∈ V 4 is a quadruple of elements of a projective plane P (see §1). We observe that Γ is a complete configuration on the set Ωx = Ξ ∪ {ξk ξl ∈ Ξ(2) : xk = xl , t1,P (xk ) = t1,P (xl )}, whence it follows that the mapping i : Ωx → V,

ξj → xj , ξk ξl → xk xl

is well defined, where xk xl is the (only) element of P incident to both xk and xl . Set tx = (i ◦ t1,P , (i, i) ◦ t2,P ). Then Γ = (Ωx , Ω2x , tx ) and i ∈ Embstr (Γ, P). We call i the embedding of Γ associated with x.

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76

S. EVDOKIMOV AND I. PONOMARENKO

§5. Counting embedding numbers s with t1 (A) = t1 (B), we set Let Γ = (Ω, E, t) be a configuration. Given A, B ∈ Ω ΓA=B = (Ω, E
, t
),

E
= E ∪ (A × B) ∪ (B × A),

t
= (t1 , t
2 ),

where t
2 (ξ, η) = t
2 (η, ξ) = e for all ξ ∈ A, η ∈ B, and t
2 (ξ, η) = t2 (ξ, η) for the other s Below we say that pairs (ξ, η) ∈ Ω2 . Let A, B, C, D be pairwise distinct elements of Ω. Q = A B C D is a quadrangle in Γ if each element of the set {A, C} is incident to each element of the set {B, D}. Lemma 5.1. Let Γ be a configuration on Ω, let Ω0 ⊂ Ω, and let i0 ∈ Embstr (Γ0 , P), where Γ0 = ΓΩ0 and P is a projective plane of order q. Suppose that the configuration Γ0 is complete. Then s \Ω s 0 such that d(A) ≤ 2, we have 1) given A ∈ Ω ⎧ 2 if d(A) = 0, ⎪ ⎪(q + q + 1) e(Γ1 , i0 , P) ⎪ ⎨(q + 1) e(Γ , i , P) if d(A) = 1, 1 0 e(Γ, i0 , P) = s 0) ⊂ Ω s 0, ⎪e(Γ1 , i0 , P) if d(A) = 2 and N (A, Ω ⎪ ⎪ ⎩ e(Γ1 , i0 , P) + q e(Γ2 , i0 , P) otherwise, where Γ1 = ΓΩ\A , Γ2 = (Γ1 )B=C with N (A) = {B, C}; s 0 , we 2) given a quadrangle Q = A B C D in Γ, not all vertices of which are in Ω have ⎧ s 0, ⎪ if {B, D} ⊂ Ω ⎨e(Γ1 , i0 , P) s e(Γ, i0 , P) = e(Γ2 , i0 , P) if {A, C} ⊂ Ω0 , ⎪ ⎩ e(Γ1 , i0 , P) + e(Γ2 , i0 , P) − e(Γ3 , i0 , P) otherwise, where Γ1 = ΓA=C , Γ2 = ΓB=D , Γ3 = (Γ1 )B=D . Proof. We observe that if p1 , p2 (respectively, l1 , l2 ) are points (respectively, lines) of the projective plane P such that pi is incident to lj for all i, j, then either p1 = p2 or l1 = l2 . So, by the definition of nonstrict embedding, we have Emb(Γ, i0 , P) = Emb(Γ1 , i0 , P) ∪ Emb(Γ2 , i0 , P), Emb(Γ3 , i0 , P) = Emb(Γ1 , i0 , P) ∩ Emb(Γ2 , i0 , P). Thus the third identity in statement 2) is obtained by the inclusion-exclusion principle. If s 0 , then the completeness of Γ0 implies that i(B) = i(D) for all i ∈ Emb(Γ, i0 , P). B, D ∈ Ω So i(A) = i(C), whence e(Γ, i0 , P) = e(Γ1 , i0 , P), which proves the first identity in statement 2). The second line in statement 2) is proved similarly. To prove statement 1), we observe that if d(A) = 0 (respectively, d(A) = 1), then each mapping i1 ∈ Emb(Γ1 , i0 , P) can be extended to i ∈ Emb(Γ, i0 , P) in q 2 + q + 1 (respectively, q + 1) ways by sending all elements of A to an arbitrary element of P of type t1 (A) (respectively, to an arbitrary element of P incident to the i1 -image of the neighbor of A). Thus, e(Γ, i0 , P) = (q 2 + q + 1) e(Γ1 , i0 , P) if d(A) = 0, and e(Γ, i0 , P) = (q + 1) e(Γ1 , i0 , P) if d(A) = 1. This proves the first two identities. The third identity follows from the completeness of the configuration Γ0 . s 0 . For j = 1, 2, denote by Embj (Γ, i0 , P) (respecSuppose d(A) = 2 and {B, C} ⊂ Ω tively, Embj (Γ1 , i0 , P)) the set of all i ∈ Emb(Γ, i0 , P) (respectively, i ∈ Emb(Γ1 , i0 , P)) such that i(B) = i(C) for j = 1, and i(B) = i(C) for j = 2. Then, obviously, Emb(Γ, i0 , P) = Emb1 (Γ, i0 , P) ∪ Emb2 (Γ, i0 , P), Emb(Γ1 , i0 , P) = Emb1 (Γ1 , i0 , P) ∪ Emb2 (Γ1 , i0 , P),

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

77

and the unions are disjoint. Set ej (Γ, i0 , P) = | Embj (Γ, i0 , P)| and ej (Γ1 , i0 , P) = | Embj (Γ1 , i0 , P)|. Since two distinct elements of the same type in a projective plane are incident to a unique element of the other type, each i1 ∈ Emb1 (Γ1 , i0 , P) extends uniquely to i ∈ Emb1 (Γ, i0 , P). So, (14)

e1 (Γ, i0 , P) = e1 (Γ1 , i0 , P).

On the other hand, each i1 ∈ Emb2 (Γ1 , i0 , P) extends to i ∈ Emb2 (Γ, i0 , P) in q + 1 ways by sending all elements of A to an arbitrary element of P incident to i1 (B) = i1 (C). Since, obviously, e2 (Γ1 , i0 , P) = e(Γ2 , i0 , P), this implies that (15)

e2 (Γ, i0 , P) = (q + 1) e2 (Γ1 , i0 , P) = e2 (Γ1 , i0 , P) + q e(Γ2 , i0 , P).

By (14) and (15), we have e(Γ, i0 , P) = e(Γ1 , i0 , P) + q e(Γ2 , i0 , P), which proves the third identity in statement 1).
Throughout the rest of the section we fix a set Ω0 and denote by Γ the class of all s 0 with configurations on sets containing Ω0 . Let Γ ∈ Γ. Given a class A of Γ not in Ω d(A) ≤ 2, we set

s 0, {Γ1 } if d(A) ≤ 1 or N (A) ⊂ Ω N (Γ, A) = {Γ1 , Γ2 } otherwise, where Γ1 = ΓΩ\A and Γ2 = ΓB=C with B and C being the neighbors of A (cf. statement 1) of Lemma 5.1). Similarly, given a quadrangle Q = A B C D in Γ, not all vertices s 0 , we set of which are in Ω ⎧ s 0, ⎪ if {B, D} ⊂ Ω ⎨{Γ1 } s 0, N (Γ, Q) = {Γ2 } if {A, C} ⊂ Ω ⎪ ⎩ {Γ1 , Γ2 , Γ3 } otherwise, where Γ1 = ΓA=C , Γ2 = ΓB=D and Γ3 = (Γ1 )B=D (cf. statement 2) of Lemma 5.1). Clearly, N (Γ, A), N (Γ, Q) are subsets of Γ. Definition 5.2. By a d-deduction in Γ we mean a rooted tree of depth d with vertices in Γ such that the set of sons of any nonleaf Γ
is equal either to N (Γ
, A) for some class s 0,Γ
with d(A) ≤ 2, or to N (Γ
, Q) for some quadrangle Q in Γ
not all A of Γ
not in Ω s 0,Γ
. If Γ is the root of this tree and a set Γ
⊂ Γ contains all vertices of which are in Ω of its leaves, we say that it is a d-deduction of Γ from Γ
. In this case for any d
≥ d we also say that Γ is d
-deducible from Γ
. This definition shows that Γ is 0-deducible from Γ
if and only if Γ ∈ Γ
. Also, if Γ is d1 -deducible from a set Γ1 , each element of which is d2 -deducible from Γ2 , then Γ is (d1 + d2 )-deducible from Γ2 . Applying Lemmas 5.1 and 4.4 and Theorem 4.3, we immediately obtain the following statement. Theorem 5.3. Suppose Γ ∈ Γ and i0 ∈ Embstr (ΓΩ0 , P), where P is a projective plane of order q. Suppose also that for some d ≥ 0 the configuration Γ is d-deducible from the set of all final configurations belonging to Γ. Then estr (Γ, i0 , P) = fΓ (q),

e(Γ, i0 , P) = gΓ (q),

where fΓ (x) and gΓ (x) are polynomials with integral coefficients independent of P and i0 .

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78

S. EVDOKIMOV AND I. PONOMARENKO

§6. Proof of Theorem 1.1 Let Γ = (Ω, E, t), where t = (t1 , t2 ), be a Ξ-configuration (see §4). It is easily seen that R(Γ, P)T = R(ΓT , P), where ΓT = (Ω, E, t
) with t
= (f1 ◦ t1 , f2 ◦ t2 ) and f1 ∈ Sym(Ω), f2 ∈ Sym(E) are the permutations induced by the involution (ξ1 , ξ3 )(ξ2 , ξ4 ) ∈ Sym(Ξ). Also,  ∆(V 2 ) = R(Γ, P), Γ

where Γ runs over all Ξ-configurations such that t2 (ξ1 , ξ3 ) = t2 (ξ2 , ξ4 ) = e. Thus, the partition Π is closed with respect to transposition, and the set ∆(V 2 ) is a union of classes of Π. To complete the proof of statement 1), it only suffices to verify that for any R, S, T ∈ Π, the number cR,S (u, w) defined by (5) does not depend on the choice of (u, w) ∈ T . We need some notation. Let Σ be the set of six distinct commuting variables ξ1 , ξ2 , ξ3 , ξ4 and σ, τ . Thus, Σ = Ξ ∪ {σ, τ }, where Ξ is the set defined in §1. Put Ω = Σ ∪ Σ(2) , where Σ(2) is the symmetric square of Σ. Obviously, |Ω| = 21 and Ω contains the set Ω0 = Ξ ∪ Ξ(2) . For k = 0, 1, 2, we introduce an injection fk : Ω0 → Ω such that ⎧ ⎪ ⎨(ξ1 , ξ2 , ξ3 , ξ4 ) if k = 0, (fk (ξ1 ), fk (ξ2 ), fk (ξ3 ), fk (ξ4 )) = (ξ1 , ξ2 , σ, τ ) if k = 1, ⎪ ⎩ (σ, τ, ξ3 , ξ4 ) if k = 2, and fk (ξi ξj ) = fk (ξi )fk (ξj ) for all i = j. Set Ωk = fk (Ω0 ) (k = 1, 2). Let Γk be a Ξ-configuration, k = 0, 1, 2. Denote by Γ
k the configuration obtained from Γk by translation of the structure along fk . Obviously, Γ
0 = Γ0 . Suppose that the configurations Γ0 , Γ1 , Γ2 are consistent, i.e., (Γ
k )Ωk ∩Ωl = (Γ
l )Ωk ∩Ωl ,

k, l = 0, 1, 2,

Γ
k .

Then there exists a uniquely determined configurawhere Ωk is the underlying set of  tion Γ on the set Ω = 2k=0 Ωk such that ΓΩk = Γ
k for all k. We denote it by [Γ0 , Γ1 , Γ2 ] and call it the initial configuration associated with Γ0 , Γ1 , Γ2 . The following statement is straightforward from the definitions. Lemma 6.1. Let P be a projective plane, let V be the set of its elements, and let Γ0 , Γ1 , Γ2 be Ξ-configurations with Γ0 = Der(x, P), where x ∈ V 4 . Suppose that the configurations Γ0 , Γ1 , Γ2 are consistent and Γ = [Γ0 , Γ1 , Γ2 ]. Then for all R = R(Γ1 , P), S = R(Γ2 , P) and u = (x1 , x2 ), w = (x3 , x4 ) we have cR,S (u, w) = estr (Γ, i0 , P), where i0 ∈ Embstr (Γ0 , P) is the embedding associated with x. We complete the proof of statement 1). Take R, S, T ∈ Π and set Γ0 = Der(T ), Γ1 = Der(R), Γ2 = Der(S). If Γ0 , Γ1 , and Γ2 are not consistent, then, obviously, cR,S (u, w) = 0 for all (u, w) ∈ T . Suppose that they are consistent and take (u, w) ∈ T . Then Γ0 = Der(x, P), where x = (u, w). By Lemma 6.1, we have cR,S (u, w) = estr (Γ, i0 , P) with Γ and i0 as in the lemma. By Theorem 6.2 below, the latter number does not depend on the choice of (u, w), which proves statement 1). Theorem 6.2. Let Γ be an initial configuration, and let i0 ∈ Embstr (Γ0 , P), where P is a projective plane of order q. Then estr (Γ, i0 , P) = fΓ (q), where fΓ (x) is a polynomial with integral coefficients not depending on P and i0 .

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Proof. This follows from Theorems 7.3 and 7.4 to be proved in the next section, in combination with Theorem 5.3.
Statement 3) of Theorem 1.1 is obtained from Theorem 6.2 as a byproduct. To prove statement 2), let Γ
∈ Der(P). Then there exists x ∈ V 4 such that Γ
= Der(x, P). Set Γ0 = Der(y, P), where y = (x1 , x2 , x1 , x2 ), Γ1 = Γ
, and Γ2 = (Γ
)T . Since, obviously, these configurations are consistent, we can form the initial configuration Γ = [Γ0 , Γ1 , Γ2 ]. Then (16)

R(Γ0 , P) = {(z1 , z2 , z1 , z2 ) ∈ V 4 : t1 (z1 ) = t1 (x1 ), t1 (z2 ) = t1 (x2 ), t2 (z1 , z2 ) = t2 (x1 , x2 )}.

Take another projective plane P
of order q
and elements x
1 , x
2 ∈ V
such that t1 (x
1 ) = t1 (x1 ), t1 (x
2 ) = t1 (x2 ), t2 (x
1 , x
2 ) = t2 (x1 , x2 ). From (16) it follows that Der(y, P) = Der(y
, P
) = Γ0 , where y
= (x
1 , x
2 , x
1 , x
2 ). We denote by i0 ∈ Embstr (Γ0 , P) and i
0 ∈ Embstr (Γ0 , P
) the embeddings associated with the quadruples y and y
, respectively. Then, by Theorem 6.2, we have estr (Γ, i0 , P) = fΓ (q),

estr (Γ, i
0 , P
) = fΓ (q
),

where q
is the order of P
. Since, obviously, estr (Γ, i0 , P) = 0, the polynomial fΓ is nonzero. Therefore, estr (Γ1 , P
) = 0 whenever q
= q or q
is larger than the maximum absolute value of a root of fΓ . In both cases we have R(Γ
, P
) = R(Γ1 , P
) = ∅, whence it follows that Γ
∈ Der(P
). This proves the first part of statement 2), and also the second part, because the number of all Ξ-configurations is finite. §7. Admissible configurations 7.1. Let Γ be an initial configuration on Ω (see §6). The definition implies that Ω ⊂ Ω and (17)

ΓΩ0 is a Ξ-configuration.

Also, Γ satisfies the following condition for ξ, η, ζ ∈ Σ: (18)

ξ, ξη ∈ Ω ⇒ t2 (ξ, ξη) = i,

and ξη, ξζ ∈ Ω ⇒ t1 (ξη) = t1 (ξζ)

(this is an easy consequence of the fact that the same is true for any Ξ-configuration on Ω and ξ, η, ζ ∈ Ξ). Moreover, conditions (17) and (18) are preserved when we pass s \Ω s 0 , as well as to the from a configuration Γ to the configuration ΓΩ\A with A ∈ Ω s s configuration ΓA=B with A, B ∈ Ω such that {A, B} ⊂ Ω0 . From now on, by a configuration we mean any configuration Γ on Ω ⊂ Ω satisfying conditions (17) and (18) (we will refer to (18) as the T -argument or the type comparison argument). We set (19)

Ω∗1 = Ω ∩ {σξ1 , σξ2 , τ ξ1 , τ ξ2 },

Ω∗2 = Ω ∩ {σξ3 , σξ4 , τ ξ3 , τ ξ4 }.

Clearly, Ω∗i = Ωi \ (Ω0 ∩ Ω3−i ), i = 1, 2. s if Definition 7.1. A configuration Γ on Ω is said to be admissible at a class A ∈ Ω ∗ ∗ either A ⊂ Ω1 ∪ Ω2 or (20)

s i, A ∩ Ω∗i = ∅ ⇒ N (A) ⊂ Ω

i = 1, 2.

The configuration Γ is said to be admissible if it is admissible at any of its classes. The next lemma follows straightforwardly from the definition.

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80

S. EVDOKIMOV AND I. PONOMARENKO

s then so are the Lemma 7.2. If a configuration Γ on Ω is admissible at a class A ∈ Ω, configurations ΓΩ\B and ΓC=D , where B, C, D are classes of Γ other than A and such s \Ω s 0 .4 that B and at least one of C, D belong to Ω Theorem 7.3. Any initial configuration is admissible. Proof. Let Γ be an initial configuration associated with the Ξ-configurations Γ0 , Γ1 , s such that A ⊂ Ω∗1 ∪Ω∗2 . and Γ2 . It suffices to verify that Γ is admissible at each class A ∈ Ω ∗ ∗ Suppose that A ∩ Ωi = ∅ for some i = 1, 2. If A ∩ Ω3−i = ∅, then we can find α0 , . . . , αk ∈ A such that t2 (αj , αj+1 ) = e for all j = 0, . . . , k − 1, where α0 ∈ Ω∗i and αk ∈ Ω∗3−i . Set l = maxαj ∈Ω∗i j. Then l < k and αl+1 ∈ Ω∗i . On the other hand, αl ∈ Ω∗i , so that the definition of an initial configuration implies that αl+1 ∈ Ωi , a contradiction. We conclude that A ⊂ Ω∗i , whence it follows that N (α) ⊂ Ωi for all α ∈ A. Thus, s i. N (A) ⊂ Ω
Let d > 0. An admissible configuration Γ on Ω is said to be d-reducible if there exists a d
-deduction of it from the admissible configurations in the class of all configurations, the underlying sets of which contain Ω0 , where 0 < d
≤ d (see Definition 5.2). In §8 we shall prove the following theorem. Theorem 7.4. Each admissible configuration other than a final one is 2-reducible. In particular, any admissible configuration is d-deducible from final ones for some d. 7.2. In this subsection we find sufficient conditions for an admissible configuration to be 1-reducible or 2-reducible (Theorem 7.7 and Lemma 7.8). For this purpose, the following definition is useful. Definition 7.5. Let Γ be an admissible configuration on Ω. We say that two classes s of the same type are comparable (respectively, quasicomparable) in Γ if A1 , A2 ∈ Ω s 0 , or the configuration ΓA =A is admissible (respectively, 1-deducible either A1 , A2 ∈ Ω 1 2 from admissible ones). Below we frequently use the following conditions sufficient for comparability and quasicomparability. s are Lemma 7.6. Let Γ be an admissible configuration on Ω. Suppose that A1 , A2 ∈ Ω s 0 . Then: classes of the same type and that one of them does not belong to Ω 1) A1 and A2 are comparable whenever either Ai ⊂ Ω∗1 ∪ Ω∗2 for some i, or N (A1 ) = N (A2 ), s 0 and Γ contains a path 2) A1 and A2 are quasicomparable whenever A1 , A2 ∈ Ω s s 0 , or N (A1 ) \ {B} = P = A1 B C D A2 with C ∈ Ω0 such that either B, D ∈ Ω N (A2 ) \ {D} and N (B) \ {A1 } = N (D) \ {A2 }. Proof. Lemma 7.2 shows that the configuration Γ
= ΓA1 =A2 is admissible at each class other than A = A1 ∪ A2 . Suppose that the hypothesis of statement 1) is fulfilled. If Ai ⊂ Ω∗1 ∪ Ω∗2 for some i, then, obviously, Γ
is admissible at A (Definition 7.1), and hence, the classes A1 and A2 are comparable. Otherwise, the neighborhood of A in Γ
coincides with N (A1 ) = N (A2 ). Since Γ is admissible at both A1 and A2 , this implies that Γ
is admissible at A (see (20)). Statement 1) is proved. Suppose that the hypothesis of statement 2) is fulfilled. Then Q = A B C D is a s 0 , the configuration Γ
1 = Γ
quadrangle in Γ
. Moreover, since C ∈ Ω A=C is admissible s 0 , then Γ
is at the class A ∪ C; hence, it is admissible by Lemma 7.2. If B, D ∈ Ω


1-deducible from Γ1 and we are done. Otherwise, Γ is 1-deducible from Γ1 , Γ
2 = Γ
B=D , and Γ
3 = (Γ
2 )A=C = (Γ
1 )B=D . Therefore, by Lemma 7.2 it suffices to verify that 4 Obviously,

A is a class of each of the configurations ΓΩ\B and ΓC=D .

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

81

the configuration Γ
2 is admissible. However, by the same lemma, this configuration is admissible at each class other than A and B ∪ D. From the hypothesis it follows that the neighborhoods of A1 and A2 in Γ

= ΓB=D coincide. Since Γ

is admissible at both A1 and A2 , this implies that the configuration Γ
2 = Γ

A1 =A2 is admissible at the class A (see the preceding paragraph). Similarly, the neighborhoods of B and D in Γ
coincide and Γ
is admissible at both B and D. Thus, the configuration Γ
2 is admissible at the class B ∪ D.
Theorem 7.7. Let Γ be an admissible configuration on Ω. Then the following two statements hold true: s Ω s 0 and d(A) ≤ 2. Suppose that the neighbors of A are comparable (re1) Let A ∈ Ω\ spectively, quasicomparable) in ΓΩ\A whenever d(A) = 2. Then Γ is 1-reducible (respectively, 2-reducible). s 0. 2) Let Q = A B C D be a quadrangle in Γ, not all vertices of which belong to Ω Suppose that A and C are comparable in Γ (respectively, quasicomparable in Γ s 0 ). Then Γ is 1-reducible and also in ΓB=D if B or D does not belong to Ω (respectively, 2-reducible) whenever B and D are comparable. Proof. We prove statement 1). Since d(A) ≤ 2, the configuration Γ is 1-deducible from the set {Γ1 } if d(A) < 2, and from the set {Γ1 , Γ2 } if d(A) = 2, where Γ1 = ΓΩ\A , and Γ2 = (Γ1 )B=C , B and C being the neighbors of A. By Lemma 7.2, the configuration Γ1 is admissible. Suppose that Γ2 enters the deduction. Then at least one of B and s 0 \ {A}. So, if the classes B and C are comparable s 0,Γ = Ω C does not belong to Ω 1 (respectively, quasicomparable) in Γ1 , then, by Definition 7.1, the configuration Γ2 is admissible (respectively, 1-deducible from admissible ones), and hence the configuration Γ is 1-reducible (respectively, 2-reducible). s 0 , the configuration Now we prove statement 2). Since not all vertices of Q belong to Ω Γ is 1-deducible from the set {Γ1 , Γ2 , Γ3 }, where Γ1 = ΓA=C , Γ2 = ΓB=D , and Γ3 = (Γ1 )B=D = (Γ2 )A=C . If Γ1 enters the deduction, then either A or C does not belong to s 0 . So, by Definition 7.5, the configuration Γ1 is admissible (respectively, 1-deducible Ω from admissible ones) whenever the classes A and C are comparable (respectively, quasicomparable) in Γ. Suppose that the classes B and D are comparable in Γ. Then if Γ2 enters the deduction, it is admissible by the above argument. Suppose Γ3 enters the deduction. Then so do Γ1 and Γ2 . Therefore, if A and C are comparable in Γ, then these configurations are admissible by the above, and hence, by Lemma 7.2, the configuration Γ3 is admissible. Finally, let A and C be quasicomparable in Γ. Since either s 0 , the hypothesis of statement 2) implies that A and C are B or D does not belong to Ω quasicomparable in Γ2 , and hence, the configuration Γ3 is 1-deducible from admissible configurations. Thus, in the comparability (respectively, quasicomparability) case, the configuration Γ is 1-deducible (respectively, 2-deducible) from admissible configurations; i.e., it is 1-reducible (respectively, 2-reducible).
The sufficient condition of 1-reducibility (respectively, 2-reducibility) occurring in statements 1) and 2) of Theorem 7.7 will be referred to as the D-argument and the Q-argument (respectively, the generalized D-argument and the generalized Q-argument). Statement 1) of Lemma 7.6 shows that two classes of a configuration Γ are comparable whenever one of them belongs to the set (21)

X ∗ = X \ ΩĞ 1 ∩ Ω2 ,

s \Ω s 0. X=Ω

In these cases we shall use the D-argument and the Q-argument without any reference to the above statement (and even without any mention that the classes are comparable).

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82

S. EVDOKIMOV AND I. PONOMARENKO

Below we shall use the following property of an admissible but not 1-reducible configuration Γ on Ω: given A ∈ X and ξ, η ∈ Ω0 , we have s ηs ∈ N (A, Ω s 0 ), ξs = ηs ⇒ {ξ, η} = {ξi ξj , ξk } or {ξ, η} = {ξi ξj , ξk ξl } (22) ξ, with distinct i, j, k, l ∈ {1, 2, 3, 4}. Indeed, if ξ = ξi , η = ξj (respectively, ξ = ξi ξj , η = s Ěs Ě ξi ξk ), then, by the Q-argument with Q = A ξsi ξĚ i ξj ξj (respectively, Q = A ξi ξj ξi ξi ξk ), the configuration Γ is 1-reducible. We observe that t1 (ξ) = t1 (η). So, by the T-argument, the set {ξ, η} cannot contain both a variable and the product of this variable by another variable. This proves (22). Lemma 7.8. Let Γ be an admissible configuration on Ω. Suppose that at least one of the following conditions is satisfied : s 0 ) ≥ 3 for some A ∈ X; 1) d(A, Ω s 0 ) ≥ 4 for some incident classes A, B ∈ X; s 0 ) + d(B, Ω 2) d(A, Ω s 0 ) + d(C, Ω s 0 ) ≥ 4 and d(A, Ω s 0 ) ≥ 1 for some distinct comparable classes 3) d(B, Ω B, C ∈ X such that A is incident to both B and C. Then the configuration Γ is 1-reducible. s ηs, ζs be pairwise distinct elements of N (A, Ω s 0 ), where ξ, η, ζ ∈ Ω0 . Since Proof. Let ξ, all 2-subsets of the set {ξ, η, ζ} cannot simultaneously be of the form (22), statement 1) s 0) = follows. We prove statement 2). By statement 1), we may assume that d(A, Ω s 0 ) = 2. From (22) (applied to A and B) it follows that ξĚ s d(B, Ω ξ ∈ N (A, Ω ) and ξĚ i j 0 k ξl ∈ s 0 ) for some i, j, k, l. Then the T-argument shows that {i, j, k, l} = {1, 2, 3, 4} and N (B, Ω s s s s 0 other than ξĚ that the neighbor of A in Ω i ξj is either ξk or ξl , say ξk . However, by the s Q-argument with Q = A B ξĚ ξ , the configuration Γ is 1-reducible. ξ k l k s 0) s 0 ) = d(C, Ω Now we prove statement 3). By statement 1), we may assume that d(B, Ω s 0 ), then the Q-argument with Q = D B A C shows that the s 0 )∩N (C, Ω = 2. If D ∈ N (B, Ω configuration Γ is 1-reducible. Thus, applying (22) to B and C and using the T-argument, s 0 ), say the first set, is of the form s 0 ), N (C, Ω we see that at least one of the sets N (B, Ω Ě s 0 ) = {ξĚ ξ , ξ ξ }, where i, j, k, l are pairwise distinct. From the hypothesis of the N (B, Ω i j k l s 0 ). The T-argument statement it follows that there exists ξ ∈ Ω0 such that ξs ∈ N (A, Ω Ě shows that ξ ∈ Ξ. Put D = ξĚ ξ if ξ ∈ {ξ , ξ }, and D = ξ ξ if ξ ∈ {ξk , ξl }. Then, by i j i j k l the Q-argument with Q = A ξsD B, the configuration Γ is 1-reducible.
§8. Proof of Theorem 7.4


Given ξ ∈ Σ, we set ξ = ξ f , where f = (ξ1 , ξ2 )(ξ3 , ξ4 )(σ, τ ) ∈ Sym(Σ); if ξ = ηζ with η, ζ ∈ Σ, then, by definition, ξ
= η
ζ
. The element ξ
is called the twin of ξ. In what follows, by convention, given a configuration Γ on Ω, we set ξs = ∅ whenever ξ ∈ Ω \ Ω. Concerning the notation X and X ∗ , see (21). Lemma 8.1. Suppose that an admissible configuration Γ is neither final nor 1-reducible. Then X ∗ = ∅ and d(A) = 3 for each A ∈ X ∗ . Moreover, exactly one of the following two statements is true: sσ Ď ∈ X ∗ , we have N (A) = {ξ, s, τs}; I) for all υ ∈ {σ, τ } and ξ ∈ Ξ such that A = υξ ∗ Ď s II) for all υ ∈ {σ, τ } and ξ ∈ Ξ such that A = υξ ∈ X , we have N (A) = {ξ, υ s, A
}, where A
is the class containing the twin of υξ. s are pairwise comparable. Since Γ is not Proof. Suppose X ∗ = ∅. Then all elements of Ω final, the set X is not empty. So, by the D-argument, without loss of generality we may s ≥ 3 for all A ∈ X. In particular, if d(A, X) = 0 for some A ∈ X, assume that d(A, Ω) then the configuration Γ is 1-reducible by statement 1) of Lemma 7.8, a contradiction. Otherwise |X| ≥ 2 and d(A, X) ≥ 1 for all A ∈ X. If |X| = 2, then Γ is 1-reducible by

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

83

statement 2) of Lemma 7.8, a contradiction. Thus, |X| = 3 and X = {s σ , τs, σĎτ }. Then Ď0 ) by the T-argument. So, the configuration Γ is 1-reducible N (Ď σ τ ) = {s σ , τs} ∪ N (Ď στ , Ω Ď0 ) = 0, and by statement 3) of Lemma 7.8 by the D-argument with A = σĎτ if d(Ď στ , Ω Ď with A = σĎτ , B = σ s, and C = τs if d(Ď σ τ , Ω0 ) > 0. Again we arrive at a contradiction. Thus, X ∗ = ∅. Ď ∈ X ∗ , where υ ∈ {σ, τ } and ξ ∈ Ξ. Then A ⊂ Ω∗ ∪ Ω∗ . Without loss of Let A = υξ 1 2 generality we may assume that υ = σ and ξ = ξ1 . Since the configuration Γ is admissible, s 1 . So, by the T-argument, we conclude that we have N (A) ⊂ Ω s, τs}. N (A) ⊂ {A
, ξs1 , ξs2 , σ If ξs1 = ξs2 and ξs2 ∈ N (A), then the configuration Γ is 1-reducible by the Q-argument with
s Q = ξs1 Aξs2 ξĚ s, τs}. By the T-argument we 1 ξ2 , a contradiction. Thus, N (A) ⊂ {A , ξ1 , σ s s conclude that either N (A) ⊂ {ξ1 , σ s, τs} or N (A) ⊂ {ξ1 , σ s, A
}. In any case the elements of N (A) are pairwise comparable. Since Γ is not 1-reducible, the D-argument implies that d(A) = 3. In particular, the neighborhood of each class in X ∗ is of type I or of type II. If there are no neighborhoods of type I, we are done. Otherwise, we have σ, τ ∈ Ω and (23)

t1 (σ) = t1 (τ ).

In this case, if there are no neighborhoods of type II, then we are done. Otherwise, there Ď ∈ X ∗ with neighborhood of type II. Since the class A
contains exists a class A = υξ

the element υ ξ , we have t1 (υ) = t1 (υ
ξ
) = t1 (υ
) (T-argument). Since {υ, υ
} = {σ, τ }, this implies that t1 (σ) = t1 (τ ), which contradicts (23). Thus, exactly one of statements I), II) is true.
Now we turn to the proof of Theorem 7.4. Let Γ be a nonfinal admissible configuration. There is no loss of generality in assuming that Γ is not 1-reducible. Then, by Lemma 8.1, we have X ∗ = ∅ and d(A) = 3 for all A ∈ X ∗ . We consider two cases, depending on which particular statement, I) or II), of that lemma holds true. Case I. In this case, σ, τ ∈ Ω and t1 (σ) = t1 (τ ). Consider two possibilities. sσ Ě Suppose |X ∗ | = 1. Then X ∗ = {A} and N (A) = {ξ, s, τs}, where A = υ 0 ξ, υ0 ∈ {σ, τ }, and ξ ∈ Ξ. So, στ ∈ Ω, because otherwise, by the Q-argument with Q = σ s A τs σĎτ , the configuration Γ is 1-reducible (στ is incident to both σ and τ by the T-argument). We s 0 ∪{A}, where υ ∈ {σ, τ }. If υ s ∈ X, have arrived at a contradiction. Therefore, N (s υ) ⊂ Ω s then d(s υ , Ω0 ) ≥ 2 by the D-argument. Thus, we get a contradiction with statement 1) of s 0 , with statement 2) if only one of these elements, say σ s, belongs Lemma 7.8 if σ s, τs ∈ Ω s 0 (B = σ s s), and with statement 3) if σ s, τs ∈ Ω s, C = τs). to Ω0 (B = σ Ě Suppose |X ∗ | ≥ 2. In X ∗ we take two distinct classes A1 = υ Ě 1 ξ and A2 = υ 2 η, s where υ1 , υ2 ∈ {σ, τ } and ξ, η ∈ Ξ. Then N (A1 ) = {ξ, σ s, τs} and N (A2 ) = {s η, σ s, τs}. If ξs = ηs, then N (A1 ) = N (A2 ). By statement 1) of Lemma 7.6, this shows that A1 and A2 are comparable. So, by the Q-argument with Q = σ s A1 τs A2 , the configuration Γ is 1-reducible. We have arrived at a contradiction. Thus, ξs = ηs. Then, by statement 2) of Ď ηs A2 , the classes A1 and A2 are quasicomparable. Now the Lemma 7.6 with P = A1 ξsξη generalized Q-argument with Q = σ s A1 τs A2 shows that the configuration Γ is 2-reducible, and we are done. Case II. In this case, the T-argument implies that τ σ ∈ Ω and if σ, τ ∈ Ω, then t1 (σ) = t1 (τ ). Furthermore, the following is true. Ď ∈ X ∗ , where υ ∈ {σ, τ } and ξ ∈ Ξ. Then: Lemma 8.2. Suppose υξ 1) υs ∈ X; Ď υĎη} for some η ∈ Ξ; 2) d(s υ ) = d(s υ , X ∗ ) = 2, and N (s υ ) = {υξ, 3) t1 (ξ) = t1 (η) = t1 (υ), t1 (ξ
) = t1 (η
) = t1 (υ).

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84

S. EVDOKIMOV AND I. PONOMARENKO

Ď By Proof. Without loss of generality we assume that υ = σ and ξ = ξ1 . Set A = υξ.

Ď s s, A } with A = τ ξ2 . In particular, t1 (ξ1 ) = Lemma 8.1, d(A) = 3 and N (A) = {ξ1 , σ s τs A
A, the classes σ t1 (ξ2 ). Moreover, if τ ∈ Ω, then, by the Q-argument with Q = σ and τ are not incident. s 0 . Then by (22) we have σ s = ξĚ Suppose that σ s ∈ Ω j ξk for distinct j, k ∈ {2, 3, 4}. Moreover, j, k = 2 by the Q-argument with Q = A σ s ξs2 A
. Thus, σ s = ξĚ 3 ξ4 . Also, by

∗ s statement 1) of Lemma 7.8 we have A ∈ Ω0 . So, A ∈ X , because otherwise A
= τs and t1 (A
) = t1 (σ) = t1 (τ ) in contrast to the T-argument. Therefore, by Lemma 8.1, we have τ ∈ Ω and τs = ξs2 . Applying statement 2) of Lemma 7.8 with B = A
, we see that s 0 . Since t1 (ξ1 ) = t1 (ξ2 ) = t1 (ξ3 ) = t1 (ξ4 ), it follows that τs ∈ Ω Ě Ě s 0 ) ⊂ {ξs1 , ξĚ N (s τ, Ω 2 ξ3 , ξ2 ξ4 , ξ3 ξ4 }. So, the Q-argument with Q = A A
τs ξs1 , Q = τs A
ξs2 ξĚ s A
ξs2 ξĚ 2 ξ3 , Q = τ 2 ξ4 , and Q =
s A A τs σ s shows that N (s τ , Ω0 ) = ∅. Since the classes σ s and τs are not incident, this implies that N (s τ ) ⊂ X ∗ , whence N (s τ ) ⊂ X ∗ ∩ {τĎ ξ2 , τĎ ξ3 , τĎ ξ4 } (T-argument). From the D-argument it follows that d(s τ ) > 1, so that τĎ ξi ∈ N (s τ ) for Ě4 ∈ N (τĎ some i ∈ {3, 4}, say i = 3. Since τĎ ξ3 ∈ X ∗ , we conclude that σξ ξ3 ) by Lemma 8.1, which contradicts the fact that t1 (ξ3 ) = t1 (ξ4 ). This proves statement 1). σ ), the D-argument with A = σ s implies Suppose that d(s σ , X ∗ ) < 1. Then, since τs ∈ N (s s 0 ) ≥ 2. Since t1 (ξ1 ) = t1 (ξ2 ), formula (22) yields N (s s 0 ) ⊃ {ξsi , ξĚ that d(s σ, Ω σ, Ω j ξk }, where i, j, k are pairwise distinct elements of {1, 2, 3, 4}, i = 1, j, k = 2, and ξsi = ξĚ j ξk . However, i = 2 (respectively, j = 1, k = 1) by the Q-argument with Q = A A
ξs2 σ s Ě s s (respectively, Q = A σ s ξĚ ξ , Q = A σ s ξ ξ ). So, i, j, k ∈ {3, 4}, a contradiction. ξ ξ 1 k 1 1 j 1 σ , X ∗ ) other than A. Then by the Thus, d(s σ , X ∗ ) ≥ 2. Therefore, there exists B ∈ N (s Ě3 , σξ Ě4 }, say B = σξ Ě3 . In this case, Lemma 8.1 implies that T-argument we have B ∈ {σξ N (B) = {B
, σ s, ξs3 } with B
= τĎ ξ4 . So, t1 (ξ1 ) = t1 (ξ3 ) = t1 (σ),

t1 (ξ2 ) = t1 (ξ4 ) = t1 (σ),

s 0) ⊂ σ, Ω which proves statement 3). This also shows that N (s σ , X ∗ ) = {A, B} and N (s Ě s s Ě s {ξ1 ξ3 , ξ2 , ξ4 } (T-argument). However, from the Q-argument with Q = A σ s ξ1 ξ3 ξ1 (res s spectively, Q = A A
ξs2 σ s, Q = B B
ξs4 σ s) it follows that ξĚ 1 ξ3 (respectively, ξ2 , ξ4 ) does s σ ) = {A, B}.
not belong to N (s σ , Ω0 ). Thus, N (s Since X ∗ = ∅, statement 2) of Lemma 8.2 shows that X ∗ contains distinct classes Ď and A2 = υĎη, where υ ∈ {σ, τ } and ξ, η ∈ Ξ, such that υ A1 = υξ s ∈ X and N (s υ) =

ξ
and N (A ) = {s sυ {A1 , A2 }. Moreover, N (A1 ) = {ξ, s, A
1 } with A
1 = υĚ η , υ s , A } 2 2 with
η
. Without loss of generality we may assume that A
2 = υĚ (24)

ξs = ηs.

Ď ηs A2 , the classes A1 and A2 (Otherwise, by statement 2) of Lemma 7.6 with P = A1 ξsξη are quasicomparable. Then by the generalized D-argument with A = υ s, the configuration Γ is 2-reducible and we are done.) To complete the proof, we consider two cases. s 0 . Applying (24) and statement 3) of Suppose X ∗ = {A1 , A2 }. Then A
1 , A
2 ∈ Ω



Ě Lemma 8.2, we see that A1 = ξ η = A2 (T-argument). So, N (A1 ) = N (A2 ), whence A1 and A2 are comparable by statement 1) of Lemma 7.6. Now the D-argument with A = υs shows that the configuration Γ is 1-reducible, a contradiction. Suppose X ∗ = {A1 , A2 }. By Lemma 8.2 (applied to an element of X ∗ other than A1 and A2 ) we have X ∗ = {A1 , A2 , A
1 , A
2 } and N (υs
) = {A
1 , A
2 }, where A
1 = A
2

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

85

and υs
∈ X. Moreover, N (A
1 ) = {ξs
, υs
, A1 } and N (A
2 ) = {ηs
, υs
, A2 }. By (24) (with A1 = A
1 and A2 = A
2 ), we may assume that ξs
= ηs
. Then s = N (A2 ) \ {A
}, N (A1 ) \ {A
1 } = {s υ , ξ} 2 N (A
) \ {A1 } = {υs
, ξs
} = N (A
) \ {A2 }. 1

2

Therefore, by statement 2) of Lemma 7.6 with P = A1 A
1 ξs
A
2 A2 , the classes A1 and A2 are quasicomparable. Thus, by the generalized D-argument with A = υ s, the configuration Γ is 2-reducible. §9. Proof of Theorem 1.2 Let Γ = (Ω, E, t) be a configuration, where t = (t1 , t2 ). Given (ξ, η) ∈ E, we set t(ξ, η) = (t1 (ξ), t2 (ξ, η)); a pair (t1 , t2 ) ∈ T, where T = T1 × T2 , is denoted by (t2 )t1 (see the notation). From now on, we assume that Γ is either the configuration corresponding to a projective plane (see the beginning of §4) or a Ξ-configuration. Then for some pairs of its elements we can consider the product of them, which is also an element of Γ. It is convenient to set the value of the function t to be ∅ if one of the arguments of t is an “undefined” product. Let V be the set of elements of a projective plane P. For i, j, k, l ∈ {1, 2, 3, 4}, we set (25)

Cylt (i, j) = {x ∈ V 4 : t(xi , xj ) = t},

(26)

Cols (i, j, k) = {x ∈ V

(27)

4

t ∈ T,

: t(xi , xj xk ) = t(xj , xk xi ) = t(xk , xi xj ) = is },

s ∈ T1 ,

4 : t(xi xj , xk xl ) = is
, t(xi , xk xl ) = t(xj , xk xl ) = e
s , Col(2) s (i, j, k, l) = {x ∈ V

t(xk , xi xj ) = t(xl , xi xj ) = e
s
},

s ∈ T1 ,

where t = (t1,P , t2,P ). We observe that Cylt (i, j) = CylR (i, j), where R equals Es (respectively, Es
, Is , Is
) for t = es (respectively, e
s , is , i
s ), s ∈ T1 (see (1), (2) and §3). Next, by the above convention, if x ∈ Cols (i, j, k), then t1 (xi ) = t1 (xj ) = t1 (xk ) = s, and the elements xi , xj , xk (and, thus, the indices i, j, k) are pairwise distinct. It is easily seen that the relation given by (26) can be rephrased by demanding that the elements xi , xj , xk be collinear.5 Similarly, if x ∈ Col(2) s (i, j, k, l), then t1 (xi ) = t1 (xj ) = s and t1 (xk ) = t1 (xl ) = s
, where s
is the type dual to s. Moreover, the elements xi , xj , xk , xl , and so the indices i, j, k, l, are pairwise distinct. In fact, the relation given by (27) can also be rephrased by demanding that the elements xi , xj , xk xl as well as the elements xk , xl , xi xj be collinear and pairwise distinct. The relations defined in (25), (26) and (27) are called the cylindricity, collinearity, and double collinearity relations of the plane P. Lemma 9.1. The derivative partition Π = Der(P) associated with the plane P coincides with the partition of V 4 generated by all cylindricity, collinearity, and double collinearity relations of the plane P, i.e., with the coarsest partition of V 4 for which any of these relations is a union of classes. Proof. Let Π
denote the second partition occurring in the lemma. It is easily seen that for any i, j, k, l ∈ {1, 2, 3, 4} and t ∈ T (respectively, s ∈ T1 ) the relation Cylt (i, j) (respectively, Cols (i, j, k), Col(2) s (i, j, k, l)) is the union of the relations R(Γ, P), where Γ runs over all configurations belonging to the classes K0 (respectively, K1 and K2 ) defined 5 Elements of P are said to be collinear if there exists an element incident to all of them; thus, concurrent lines are also called collinear here.

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86

S. EVDOKIMOV AND I. PONOMARENKO

as follows: K0 = {Γ ∈ K : t(ξi , ξj ) = t}, K1 = {Γ ∈ K : t(ξi , ξj ξk ) = t(ξj , ξk ξi ) = t(ξk , ξi ξj ) = is },

(28)

K2 = {Γ ∈ K : t(ξi ξj , ξk ξl ) = is
, t(ξi , ξk ξl ) = t(ξj , ξk ξl ) = e
s , t(ξk , ξi ξj ) = t(ξl , ξi ξj ) = e
s
};

here K is the class of all Ξ-configurations. Thus, Π is coarser than Π
. Conversely, let R ∈ Π. Then R = R(Γ, P) for some Ξ-configuration Γ = (Ω, Ω2 , t). It is easily seen that  R= Rξ,η,t(ξ,η) , ξ,η∈Ω

where for t ∈ T and ξ, η ∈ Ω we have set Rξ,η,t = {x ∈ V 4 : ξ, η ∈ Ωx , tx (ξ, η) = t} with Ωx and tx such that Der(x, P) = (Ωx , Ω2x , tx ) (see the end of §4). So, it suffices to verify that any relation Rξ,η,t is a union of elements of the partition Π
. Obviously, if ξ = ξi and η = ξj for some i, j ∈ {1, . . . , 4}, then Rξ,η,t = Cylt (i, j). Let ξ = ξi and η = ξj ξk for some i, j, k ∈ {1, . . . , 4} (the case where ξ = ξj ξk and η = ξi is treated similarly). Then ⎧ ⎪ Cylis (i, j) ∩ Cylis (i, k) ∩ Cyle

(j, k) if t = es , ⎪ ⎪ s ⎪ ⎨(Cyl
(i, j) ∪ Cyl
(i, k)) ∩ Cyl
(j, k) if t = e
s , is is es
Rξ,η,t = ⎪ ((Cyles (i, j) ∪ Cyles (i, k)) ∩ Cyle
s (j, k)) ∪ S if t = is , ⎪ ⎪ ⎪ ⎩(Cyl
(i, j) ∩ Cyl
(i, k) ∩ Cyl
(j, k)) \ S if t = i
, es

es

es

s

where s ∈ T1 and S = Cols (i, j, k) if i, j, k are pairwise distinct, Finally, if ξ = ξi ξj and η = ξk ξl , then ⎧ ⎪ Rξi ,η,is
∩ Rξj ,η,is
if ⎪ ⎪ ⎪ ⎨(Cyl
(i, j) ∩ Cyl
(i, k)) \ Rξ,η,e if es
es s Rξ,η,t = (2) ⎪ Colt (i, j, k, l) ∪ S1 ∪ S2 ∪ S3 ∪ S4 if ⎪ ⎪ ⎪ ⎩(Cyl
(i, j) ∩ Cyl
(i, k)) \ R if es

es


ξ,η,is

and S = ∅ otherwise. t = es , t = e
s , t = is , t = i
s ,

where s ∈ T1 and S1 = Rξi ,η,es ∩ Cyle
s (i, j), S2 = Rξj ,η,es ∩ Cyle
s (i, j), S3 = Rξ,ξk ,es
∩ Cyle

(k, l), S4 = Rξ,ξl ,es
∩ Cyle

(k, l).
s

s

We turn to the proof of Theorem 1.2. Let C
= (V 2 , Π) be the derivative scheme of P. It is easily seen that  ∆(∆2 ) = (Cyles (1, 2) ∩ Cyles (1, 3) ∩ Cyles (1, 4)), R ⊗ S = CylR (1, 3) ∩ CylS (2, 4), s∈T1

where ∆2 = ∆(V ) and R, S are basis relations of the scheme C. Therefore, by Lemma 9.1 and the definition of the 2-extension (see (10)), we have p (29) C
≥ C. By Lemma 9.1, to prove the reverse inclusion it suffices to verify that all cylindricity, collinearity, and double collinearity relations of the plane P are relations of the scheme p We note that the statement on the cylindricity relations follows from Theorem 3.2. C. To prove the statement about the collinearity relations, suppose i, j, k ∈ {1, . . . , 4} are pairwise distinct and s ∈ T1 . We observe that Cols (i, j, k) does not change under any permutation of i, j, k. So, we can assume that {i, j} = {1, 2} or {i, j} = {3, 4}. On the p T = R∗ (C). p Then other hand, Cols (3, 4, k)T = Cols (1, 2, k
), where k
= k + 2, and R∗ (C)

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

87

without loss of generality we may assume that {i, j} = {1, 2}. However, in this case we have Cols (i, j, k) = (S1 ∩ S2 ) · Cyles (k − 2, k), where S1 is the intersection of all relations Cyle
s (a, b) with distinct a, b ∈ {1, 2, k} and S2 is the intersection of all relations Cylis (a, k
) with a ∈ {1, 2, k}, and k
is uniquely determined by the condition {k, k
} = {3, 4}. Thus, the required statement follows from p the already proved statement that any cylindricity relation is a relation of the scheme C. To prove the statement about the double collinearity relations, suppose {i, j, k, l} = {1, 2, 3, 4} and s ∈ T1 . We note that Col(2) s (i, j, k, l) does not change under any permutation of the numbers i, j as well as the numbers k, l. This implies that if {i, j}∩{1, 2} = ∅, then either {i, j} = {1, 2}, or without loss of generality we may assume that {i, k} = T

(2)

= Cols
(k, l, i, j). {1, 2}. On the other hand, if {i, j} = {3, 4}, then Col(2) s (i, j, k, l) ∗ pT ∗ p Therefore, since R (C) = R (C), in any case we can assume that {i, j} = {1, 2} or {i, k} = {1, 2}. But then a straightforward check shows that

(S1 · S2 ) ∩ Cyle
s (i, j) ∩ Cyle

(k, l) if {i, j} = {1, 2}, (2) s Cols (i, j, k, l) = (S3 · S4 ) ∩ Cyle
s (i, j) ∩ Cyle

(k, l) if {i, k} = {1, 2}, s

where S1 = Cylis (i, 3) ∩ Cylis (j, 3) ∩ Cylis (4, 3),

S2 = Cylis (2, k) ∩ Cylis (2, l) ∩ Cylis (2, 1),

S3 = Cylis (i, 3) ∩ Cylis (4, k) ∩ Cylis (4, 3), S4 = Cylis (j, 1) ∩ Cylis (2, l) ∩ Cylis (2, 1). Now the statement about double collinearity relations follows from the fact that any p cylindricity relation is a relation of the scheme C.
p Thus, we have C ≥ C. Recalling (29), we see that statement 1) of the theorem is proved. To prove statement 2), let R ∈ R(C). Then, obviously, the configuration Γ = Der(xu,v , P) does not depend on the choice of (u, v) ∈ R, where xu,v = (u, u, v, v), and Rδ = {xu,v : (u, v) ∈ R} = R(Γ, P) p (see (11)). However, statement 1) implies that R(Γ, P) is a basis relation of the scheme C. Statement 2) is proved. §10. Proof of Theorem 1.4 We deduce Theorem 1.4 from the following result to be used also in §11. Theorem 10.1. Let C and D (respectively, C
and D
) be the nonhomogeneous and homogeneous schemes of a projective plane P (respectively, P
), and let ψ0 : C → C
and ψ : D → D
be the canonical similarities. Suppose that a nontrivial similarity ϕ of C is an m-similarity for some positive integer m. Then p=D p (m) , and Φ p = {1 p, ϕ} p = CpΦp , where Cp = Cp(m) , D p=ϕ p(m) , 1) D C p with ϕ

2) ψ0 ∈ Simm (C, C ) if and only if ψ ∈ Simm (D, D ). p Since, obviously, the group Φ p leaves fixed the set Proof. It is easily seen that Cp ≥ D. m ∆m and each basis relation of D , it follows that (30)

p p ≤ CpΦp ≤ C. D

Furthermore, the scheme D contains the equivalence relation Vp2 ∪Vl2 . In view of Theorem p and i, j ∈ {1, . . . , m}, then either t1 (xi ) = t1 (xj ) for all x ∈ R, or 3.2, if R ∈ R(D)

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88

S. EVDOKIMOV AND I. PONOMARENKO

m p t1 (xi ) = t1 (xj ) for all x ∈ R. So, for any s ∈ Tm the scheme D 1 and any δ ∈ {0, 1} contains the set Xs ∪ Xs
, where

Xs = {x ∈ V m : t1 (xi ) = si , i = 1, . . . , m}

(31)

p contains and s
is obtained from s by replacing si with its dual s
i for all i, and also D the relation  (32) Rδ = {(x, y) : δxi ,yi = δi , i = 1, . . . , m} = Xs × Xss , s∈Tm 1

where ss ∈ Tm si ,si = δi for all i (the symbol δ in the expressions δxi ,yi and 1 is such that δs δssi ,si stands for the Kronecker delta). Since the relation E = Rδ0 with δ0 = (0, . . . , 0) is an equivalence relation on V m with classes Xs , and the degree of the relation (Rδ )V m /E pV m /E is semiregular. Applying Theoequals 1 for all δ, this implies that the scheme D p we conclude that rem 2.1 with C = D, p = (D p E )Ψ , D

(33)

pV m /E where Ψ is as in the theorem. In our case, the homogeneity sets of the scheme D m are of the form {Xs , Xs
}, where s ∈ T1 . So |Ψ| = 2. p contains any relation of the form m (Ri ∪ Rϕ ), where Ri ∈ R(C) for Clearly, D i i=1 all i. Moreover, by (31) we have   m m

(34) Ri = (Ri ∪ Riϕ ) ∩ (Xs × Xss ), i=1

i=1

pE ≥ C m . where s and s s are determined uniquely from the condition Ri ⊂ Vsi × Vssi . So, D pE also contains ∆m , we conclude that Recalling that D p pE = C. D

(35)

p = CpΨ ≤ CpΦp . Since the groups Ψ and Φ p are of order 2 Thus, by (30) and (33), we have D p without fixed points, this implies that Ψ = Φ, p whence statement 1) and act on R(C) follows. x0 = ψ x0 (m) . Then, given R, S ∈ To prove statement 2), let ψ0 ∈ Simm (C, C
). Set ψ R(C), we have  m

 ψx0 (Ri ∪

Riϕ )

=

i=1

m

(Riψ0 ∪ Riϕψ0 ) =

i=1




m




(Riψ0 ∪ Riψ0 ϕ ),

i=1

where ϕ is the nontrivial similarity of C (we have used the obvious relation ϕψ0 = ψ0 ϕ
). x x0 to R(D) p is the the m-extension of ψ. Thus, Since also (∆m )ψ0 = ∆
m , the restriction of ψ

ψ ∈ Simm (D, D ). Conversely, let ψ ∈ Simm (D, D ). Then Theorem 3.2 shows that (36)




(Xs ∪ Xs
)ψ = Xs
∪ Xs

, p

(Rδ )ψ = Rδ
, p

m s ∈ Tm 1 , δ ∈ {0, 1} ,

where ψp = ψp(m) and Xs
and Rδ
are defined by formulas (31) and (32) with P replaced by P
. We observe that the semiregular scheme DV m /E is generated by its homogeneity sets {Xs , Xs
} and the relations (Rδ )V m /E with full support. Therefore, by (34) and (36), the similarity ψpV m /E is induced by a bijection f : Xs → Xs
. So, by statement 4)
of Theorem 2.1, the similarity ψp can be extended to a similarity ψpE : DE → DE
such

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

89

that (ψpE )V m /E is induced by f . Then from (34) it follows that  m

ψpE Ri

=

 m

i=1

ψpE (Ri ∪

Riϕ )

∩ (Xs × Xss )ψE p

i=1

=

m

(Ri ∪ Riϕ )ψ ∩ (Xsf × Xssf )

i=1

=

m

(Ri ∪ Riϕ )ψ0 ∩ (Xs
× Xss
)

i=1

=

m




(Riψ0 ∪ Riψ0 ϕ ) ∩ (Xs
× Xss
)

i=1

=

m




(Ri
∪ (Ri
)ϕ ) ∩ (Xs
× Xss
)

i=1

=

m

Ri
.

i=1 p p D

= Cp
(see (35)). Thus, ψpE is the Moreover, obviously (∆m )ψE = ∆
m , and DE = C, E

m-extension of ψ0 , whence ψ0 ∈ Simm (C, C ).

Now statements 1) and 3) of Theorem 1.4 follow from Theorem 1.3 and Theorem 10.1 with m = 2, whereas statement 2) is a consequence of statement 1) and the 2-closedness of the scheme C (statement 2) of Theorem 1.2). §11. Proof of Theorem 1.7 For a projective plane P with element set V and s ∈ T1 , we define a binary relation on V 3 as follows: (37) Cs = {(x, y) ∈ Ts × Ts : ∃v ∈ V : t1 (v) = s and v, xi , yi are collinear, i = 1, 2, 3}, where Ts ⊂ V 3 consists of all triples of noncollinear elements of type s (see the footnote on page 85). Treating the elements of Tp and Tl as triangles in P, we see that Cp and Cl consist of all pairs of triangles that are in perspective from some point and from some line, respectively. Thus, P is a Desarguesian plane6 if and only if Cp = E · Cl · E T ,

(38) where (39)

E = {(x, y) ∈ Tp × Tl : y1 = x2 x3 , y2 = x3 x1 , y3 = x1 x2 }.

We observe that the finite Desarguesian projective planes are exactly the Galois planes [14, §6]. On the other hand, it is easily seen that (40) (41)

Cs = R · R T ,   E=

∆(Ts ) = ∆(Vs3 ) \ (Ss · SsT ),  Cylis (i, j + 3) ∩ (Tp × Tl ),

1≤i=j≤3

6 A Desarguesian plane is a plane for which the Desargues theorem holds true; for the details, see [14, p. 109].

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90

S. EVDOKIMOV AND I. PONOMARENKO

where R=

 

 Cylis (i, i + 3) ∩ (Ts × (Vs3 \ Ts )),

1≤i≤3

∆(Vs3 )

=



Cyles (i, i + 3),

1≤i≤3

Ss = Cylis (1, 4) ∩ Cylis (2, 4) ∩ Cyles (3, 4). From now on we assume that P is a Galois plane. Then a nontrivial similarity of the nonhomogeneous scheme C of P is induced by any polarity of this plane. So, it is an m-similarity for all m ≥ 1. Now, let ϕ : C → C
be a 3-similarity. Then without loss of generality we may assume that C
is the scheme of a projective plane P
and that the similarity ϕ is canonical. Theorem 3.2 shows that Cs and E (respectively, Cs
and E
) are relations of the (3) scheme Cp(3) (respectively, Cp
), where Cs
and E
are the binary relations defined for P
by (37) and (39), respectively. Moreover, the same theorem implies that, given s ∈ {p, l}, we have (Cs )ϕp = Cs
, E ϕp = E
, where ϕ p=ϕ p(3) . On the other hand, since P is a Galois plane, (38) is fulfilled in C, and
hence in C . Thus P
is also a Galois plane. Since P and P
are of the same order, they are isomorphic. Since the similarity ϕ is canonical, it is induced by any isomorphism of these planes. Thus, Sim3 (C, C
) = Sim∞ (C, C
), whence s(C) ≤ 3. Let D be the homogeneous scheme of the plane P and ψ : D → D
a 3-similarity. Without loss of generality we assume that D
is a homogeneous scheme of a projective plane P
and that the similarity ψ is canonical. Since the nontrivial similarity of C is a 3similarity (see above), Theorem 10.1 with m = 3 implies that so is the canonical similarity ψ0 : C → C
, where C and C
are nonhomogeneous schemes of P and P
, respectively. But by the first part of the same theorem, ψ0 is induced by some isomorphism. Clearly, it also induces ψ. Appendix In the following table we enumerate all Ξ-configurations Γ belonging to the set Der(P), where P is a projective plane of order q. Any such configuration satisfies the following conditions: (a) any two vertices of Γ belonging to the same class are equal (the type of the corresponding edge is e); (b) any two vertices of Γ belonging to incident classes are incident; (c) there are no quadrangles in Γ (see §§4 and 5). We represent a configuration as a bipartite graph, the parts of which consist of classes of the same type, and the edges correspond to incident classes (3rd column); the restriction of Γ to Ξ is represented in the 2nd column, whereas its informal description is in the 4th column. The label of a class is the cardinality of it; we omit the label if the class is a singleton. In the 5th column of the table we write the number of Ξ-configurations represented by the graph in the corresponding row; the number in the 6th column shows that the corresponding row applies to a projective plane of order greater than or equal to that number.

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

N

ΓΞ

Γ

•

•

Comment 4 equal elements of the same type

1 4

3

44 44 4

•

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4

3

•

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•

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3 2

2 pairs of equal elements of distinct types that are not incident

3 2

2

•

8 •

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•

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•

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•

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•

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1

1

G 44GG 44 GG 4 GG

3 collinear elements in general position; the 4th equals one of them

12

2

•



• •

2 distinct elements of the same type not incident to 2 equal elements of the other type

12

2

•



• •

2

3 elements with 2 of them being equal are not incident to the 4th

24

2

2

•





2 distinct elements of the same type, one of which is incident to 2 equal elements of the other type

24

1

•



• •

2

2 incident elements, and 2 equal elements in general position

24

1

44 •

44

4

2 nonincident elements and 2 equal elements incident to one of them

24

1

44 44 4

2 equal elements of the same type are incident to 2 distinct elements of the other type

12

1

44 44 4

2 incident elements, one of which is incident to 2 equal elements

24

2

• 2

•

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• • •

13

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•

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3 elements of the same type in general position; the 4th equals one of them

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•

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•

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2 pairs of equal elements that are incident

2

•

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•

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3 equal elements incident to the 4th

7

14

8

2

3 2

•

1

3 equal elements not incident to the 4th

6

•

6

2

•

2

•

1

2 pairs of equal elements of the same type

5

•

8

•

2

3 2

•

1

3

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•

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•

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q

4 elements of the same type, 3 of which are equal

•

•

n

4

2 •

91

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44 44 4

• 2

44 44 4

•

44 44 4

• 2

•

• 3

•

2 2

• 2

• • 3

•

• •

• 2

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92

S. EVDOKIMOV AND I. PONOMARENKO

N

ΓΞ

Γ

44 •44ww•44ww •ooo •oowow• 4 4 ww44 wwo4 o4 o o o oww

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44 44 4

G 44GG 44 GG 4 GG •

•

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26

27

28

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•

29

8

3

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www www



w •ww •

•w 3 • •4

444

4



3 elements of the same type in general position, one of which is incident to the 4th 3 nonequal collinear elements, one of which is incident to the 4th

24

2

24

2

44 •44 •

4 4 4

4

4

4

3 elements of the same type in general position, 2 of which are incident to the 4th 1 element incident to 3 distinct elements

24

1

8

2

6

3

6

2

•

•

•

• 4

• •

•

•

G 44GG 44 GG 4 GG •

•

GG •44 w• GGww4

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ww G G 4G4 w w

2 elements of the same type and 2 elements of the other type in general position double collinearity

2 incident elements and 2 nonincident elements in general position

24

2

G 44GG•44 44 GG44 4 GG4

2 nonequal elements incident to the 3rd element and not incident to the 4th

24

2

GG •44 w• GGww4

ww G G 4G4 w • •w • 2 4 •4 • 44 4

two pairs of 2 incident elements in general position

12

1

24

1

• 2

•

•

•

•

•

•

•

•

•

•

•

•

•

44 • 44 4

3

3 nonequal collinear elements not incident to the 4th

•

•

31

2

2

•

30

2

8

2

•

8

3 elements of the same type in general position not incident to the 4th

•

•

2

• ww• ww •

www www



w •ww •

•w 3 •

•444

44

•

22

2

4 nonequal collinear elements

•

20

q

O G 44GOGOO 44 GGOOO 4 GG OOO • • • • 6

•

4 nonequal elements of the same type, 3 of which are collinear

n

•

•

19

Comment 4 elements of the same type in general position

•

•

•

•

•

• 2

2 incident elements, one of which is incident to the 3rd element and another one to the 4th

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SCHEMES OF A FINITE PROJECTIVE PLANE AND THEIR EXTENSIONS

93

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Received 18/APR/2008 Translated by THE AUTHORS

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