Circle Spaces in a Linear Space

Transcription

1 Circle Spaces in a Linear Space Sandro Rajola and Maria Scafati Tallini Dipartimento di Matematica, Universitá di Roma La Sapienza, Piazzale Aldo Moro 2, ROMA, Italy. Abstract We study the circle spaces such that the circles are remarkable sets of a linear space, such as m-arcs and Fano planes. Mathematics Subject Classification (2000): Geometric structures. Keywords: Circle spaces, Steiner systems, Moebius planes. 1 Introduction A linear space is a pair S = (S, L), where S is a non-empty set whose elements we call points and L is a family of subsets of S whose elements we call lines, such that every line contains at least two points and through two points there is a unique line. A circle space (S, L, C) is a linear space (S, L) together with a subset C of the power set of S such that through three non-collinear points there is a unique element of C called circle. The aim of this research is to consider circle spaces where the circles are remarkable sets of a linear space. Here in particular we consider m-arcs and Fano planes. First we investigate the circle spaces such that the circles are m-arcs. We call them C m spaces. We prove that in an affine plane over a field k, also infinite, C m spaces exist, if m = 4. Moreover, we characterize the finite projective planes of order p, π p, p a prime, which contain a C m space, m > 3. We consider also circle spaces where the circles are Fano planes and we prove that in π q, q odd, such circle spaces do not exist. 1

2 2 2 C m Spaces in a Steiner System Let (S, L, C) be a circle space. An m-arc of S is a set of m three by three noncollinear points. Let m be an integer, m 3. We call C m space a circle space such that every circle is an m-arc. A Steiner system S(2, k, v) is a linear space (S, L) such that l = k, l L and S = v. Let (S, L, C) be a circle space. If there is another family R of subsets of S such that: 1. (S, L, R) is also a circle space, and 2. R R, C C : R C, then R is called a refinement of C. Let (S, L, C) be a C m space. Let R be a refinement of C, whose circles have all the same size. For every C C, set R(C) := {R R R C}. Then (C, R(C)) is a Moebius structure [1], i. e., (M1): a, b, c ( C 3) =!R R(C) : a, b, c R and (M2): R 1, R 2 R(C) it is R 1 = R 2 are satisfied. We remark that if r = R, R R, and m = C, then R(C) = [m(m 1)(m 2)]/[r(r 1)(r 2)] is an integer. Let (S, L, C) be any circle space. If one of the circles C C can be endowed with a Moebius structure (C, M) and all the circles of C have the same size m (that is (M1), (M2) are satisfied), then there is a refinement R = C C R(C) of C such that (C, R(C)) = (C, M). Indeed we can transfer the structure M of C upon any other circle C C with an arbitrary bijection from C onto C. If q is a prime power and either m = q 2, or m = q 2 + q + 1 respectively, the circle C can be provided with a structure G such that (C, G) is a desarguesian affine respectively projective plane. Theorem 1 Let (S, L, C) be a C m space where (S, L) = S(2, k, v) is a Steiner system. Then λ 2 := (v k)/(m 2) is an integer, (1) C = [v(v 1)(v k)]/[m(m 1)(m 2)] is an integer, (2) C(X) = [(v 1)(v k)]/[(m 1)(m 2)] is an integer, (3) where λ 2 is the number of circles through two distinct points of S, X S and C(X) := {C C : X C}.

3 3 Proof: Let X and Y be two distinct points of S, let r be the line XY and let C(X, Y ) := {C C, X, Y C}. Then to each point Z S r there is exactly one C C(X, Y ) passing through Z. Therefore C (X, Y ) := {C {X, Y }, C C(X, Y )} is a partition of S r. Consequently S r = v k = C(X, Y ) (m 2) and so λ 2 := C(X, Y ) = (v k)/(m 2) is an integer. If we count in different ways the incidences between pairs of points and circles, we obtain ( ) v 2 λ2 = C ( ) m 2, i. e., C = [v(v 1)(v k)]/[m(m 1)(m 2)] is an integer. If X S is fixed, then each C C(X) is incident with m 1 points of S {X} and each point Y S {X} is incident with λ 2 circles of C(X). Therefore C(X) (m 1) = (v 1)λ 2 = [(v 1)(v k)]/(m 2) and (3) is satisfied. From Theorem 1 it follows immediately: Corollary 2 If α q is a finite affine plane of order q, then the existence of a C 4 and of a C q+1 space is arithmetically possible for any q. Corollary 3 If π q is a finite projective plane of order q and if there is a C m space, with m even, then q is even. Theorem 4 If in π q, q a prime, a C m, m > 3, exists, then q = 2 and m = 4. Proof: By (1), we get (v k)/(m 2) = p 2 /(m 2) and by m > 3 it follows either m 2 = p, or m 2 = p 2. The last case is impossible, otherwise we get m = p > p + 2, a contradiction, since for any m-arc it is m p + 2. Then m = p + 2 and p is an even prime, hence p = 2 and m = 4. Theorem 5 Let α q be a finite affine plane of order q > 3. If a C q space exists, then q = 4. Proof: From m 2 = q 2, v k = q(q 1), by (1) we get that (q 2 q)/(q 2) = q /(q 2) is an integer. Since q > 3, it follows q = 4. Similarly we prove the following Theorems. Theorem 6 Let α q be a finite affine plane of order q. If q > 4 and if in α q a C q 1 exists, then q = 5. If q > 5, then C q t spaces, with t = 0, 1, 2 do not exist. Theorem 7 Let π q be a finite projective plane of order q. Then 1. If q is even and if in π q a C q+2 space exists, then either q = 2, or q = In π q a C m does not exist for the following pairs of values of m and q: m = q+1, q 3; m = q, q 4; m = q 1, q 5; m = q 2 and m = q 3, q 7; m = q 5, q If q 8 and a C q 4 exists, then q = 8.

4 4 Proof: By m 2 = q, v k = q 2 + q + 1 (q + 1) = q 2, from (2) we get that q 3 q 2 + 3(q 2) + 12/(q + 2) is an integer. It follows either q + 2 = 4, or q + 2 = 6, or q + 2 = 12 and therefore either q = 2, or q = 4, since the plane π 10 does not exist. Similarly we prove the two further statements of the Theorem. Corollary 8 In π q, q 7, neither C q+t spaces, t = 0, 1, 2, nor C q t spaces, t = 1, 2, 3 exist. 3 Examples of C m Spaces in Affine and Projective Planes EXAMPLE 1. It is easy to check that in the Fano plane P G(2, 2) the 4-arcs are circles. Therefore we get a C 4 space in P G(2, 2). EXAMPLE 2. Let (L, k) be a separable quadratic field extension, the involutory k-automorphism, (S, L) := E(L, k) the corresponding Euclidean plane (see [1], page 186) and C := {C a,b a, b L, b 0} with C a,b := {z L (z a)( z ā) = b b} the family of all circles. Then (S, L, C) is a circle space C k +1. In particular for every prime power q there is such an example of C q+1 space. EXAMPLE 3. Let (M, M) be a Moebius plane (see [1], page 95, 96), let P M be fixed, (M, M) P = (S, L), with S := M {P }, L := {C {P }, C M(P )} the derivation of (M, M) in the point P (which is an affine plane) and C := M M(P ). Then (S, L, C) is a circle space. EXAMPLE 4. In any affine plane α k over a field k of characteristic 2, the family C of all parallelograms is a C 4 space. From EXAMPLE 4 it follows: Corollary 9 In AG(2, q), q even, C 4 spaces exist. EXAMPLE 5. In the Euclidean plane E(L, k) let: the set of all rectangular triangles (a, b, c), where a, c is the hypotenuse, the set of all rectangles (a, b, c, d), where a, c and b, d are the diagonals, the set of non-rectanguar triangles, the set of all quadrilaterals (a, b, c, d) with (a, b, c) and d the orthocenter of (a, b, c). Then: (a, b, c),!d L: (a, b, c, d), (a, b, c),!d L: (a, b, c, d),

5 5 and if (a, b, c, d), then (b, c, d, a). Thus we have: Theorem 10 The triple (L, L, Q), where Q = is a C 4 space. 4 Circle Spaces where the Circles are Fano Planes Theorem 11 In π q, q odd, circle spaces, where the circles are Fano planes do not exist. Proof: If such a circle space F exists, in π q, q odd, for any F j F, let A be the family of the 4-arcs contained in the Fano planes of F. Let π q = (S, L). Then (S, L, A) is a C 4 space: a contradiction, since Corollary 3 says that q must be even. References [1] KARZEL, H. and KROLL, H. J., Geschichte der Geometrie seit Hilbert, Wissenschaftliche Buchgesellshaft, Darmstadt, [2] SCAFATI, M. and TALLINI, G., Geometrie di Galois e Teoria dei Codici, Ed.CISU, Roma, [3] TALLINI, G., Lezioni di Geometria III, a.a Teoria dei c-insiemi in uno spazio di Galois. Blocking sets in P G(r, q) ed in AG(r, q), Dipartimento di Matematica Univ.Roma La Sapienza. [4] TALLINI, G., Lezioni di Geometria IV, a.a Disegni e spazi lineari, Dipartimento di Matematica Univ. Roma La Sapienza.