Des. Codes Cryptogr. (2009) 51:157–165 DOI 10.1007/s10623-008-9251-1
A hemisystem of a nonclassical generalised quadrangle John Bamberg · Frank De Clerck · Nicola Durante
Received: 11 September 2008 / Revised: 28 October 2008 / Accepted: 28 October 2008 / Published online: 4 December 2008 © Springer Science+Business Media, LLC 2008
Abstract The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points H such that every line meets H in half of the points of . If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q 2 ) were those of the elliptic quadric Q− (5, q), q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 52 ), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3 · A7 -hemisystem of Q− (5, 5), first constructed by Cossidente and Penttila. Keywords Hemisystem · Partial quadrangle · Strongly regular graph · Association scheme Mathematics Subject Classifications (2000)
Primary 05B25 · 05E30 · 51E12
1 Introduction A partial quadrangle PQ(s, t, µ), introduced by Cameron [3], is a geometry of points and lines such that every two points are on at most one line, there are no triangles, every line has
Communicated by G. Lunardon. J. Bamberg (B) · F. De Clerck Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281–S22, 9000 Ghent, Belgium e-mail:
[email protected] F. De Clerck e-mail:
[email protected] N. Durante Dipartimento di Matematica ed Applicazioni, Università di Napoli “Federico II”, 80125 Naples, Italy e-mail:
[email protected]
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the same number s + 1 of points, every point is incident with the same number t + 1 of lines, and there is a constant µ such that for every pair of noncollinear points there are precisely µ common neighbours. The point graph of this geometry is strongly regular. Generalised quadrangles are partial quadrangles with µ = s(t + 1), that is, they satisfy the condition that for every point P and line which are not incident, there is a unique point of collinear with P. The only known partial quadrangles, which are not generalised quadrangles, are thin (i.e., they have two points on every line), one of three exceptional examples or arise from removing points from a generalised quadrangle of order (s, s 2 ) (see [6]). To be more precise, let G be a generalised quadrangle of order (s, s 2 ) and let P be a point of G . Then by removing all those points which are collinear with P results in a partial quadrangle PQ(s − 1, s 2 , s(s − 1)) (see [4]). The only other known construction of a (thick) partial quadrangle relates to a hemisystem of G . In 1965, Segre [12] proved in his work on regular systems of the Hermitian surface, that an m-ovoid of Q− (5, q) exists only if q is odd and m = (q + 1)/2; that is, it is a hemisystem. An m-ovoid of a generalised quadrangle is a set of points such that each line meets it in m points. Segre proved that for q = 3, there is just one hemisystem up to equivalence, and it was long thought to be the only example of such an object. However, Cossidente and Penttila [5] constructed an infinite family of hemisystems of Q− (5, q) each admitting P− (4, q), together with a special example for q = 5 whose stabiliser in PU(4, 52 ) is (3 · A7 ).2, but which reduces to 3 · A7 when intersected with PGU(4, 52 ). Until their work, it was not known, nor believed, that an infinite family of hemisystems of Q− (5, q) existed. It can be found implicitly in the work of Cameron et al.[4, Proposition 2.2], that a hemisystem of a generalised quadrangle of order (s, s 2 ) gives rise to a partial quadrangle PQ((s −1)/2, s 2 , (s −1)2 /2). The partial quadrangle that we construct in this paper arises from a hemisystem of a Fisher–Thas–Walker–Kantor generalised quadrangle, and we will show that it is a new example. A partial quadrangle PQ((q − 1)/2, q 2 , (q − 1)2 /2), q 5, fully embedded in Q− (5, q) necessarily gives rise to a hemisystem, and our partial quadrangle is not a hemisystem of Q− (5, 5) (see Lemma 5.1). Alternatively, the automorphism group of our partial quadrangle is solvable and so readily distinguishable from the examples of Cossidente and Penttila. Main Theorem There is a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 52 ) with automorphism group AGL(1, 5) × S3 . The partial quadrangle PQ(2, 25, 8) has an associated strongly regular graph with parameters (v, k, λ, µ) = (378, 52, 1, 8). According to [5, Remark 4.3], the parameters of the strongly regular graph above were new at the time of the construction of their 3 · A7 -hemisystem, and hence we have constructed a second strongly regular graph with these parameters, that is inequivalent to the one known before. We also give a new construction of the 3 · A7 -hemisystem of Q− (5, 5) (Sect. 5). Both the construction of the hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle in the Main Theorem above, and the 3 · A7 -hemisystem of Q− (5, 5), arise from a certain partition of the Hermitian unital H(2, 52 ). The two generalised quadrangles of central importance in this paper are the Hermitian surface H(3, q 2 ) of order (q 2 , q) and its dual forming the points and lines of the elliptic quadric Q− (5, q). The theory of generalised quadrangles is quite diverse, and we will only need to remind the reader of those generalised quadrangles arising from flocks of quadratic cones. A flock of the quadratic cone C with vertex X in PG(3, q) is a partition of the points of C \{X } into conics. Thas [13] showed that a flock gives rise to a generalised quadrangle of order (q 2 , q), which we call a flock quadrangle. If the flock is linear, that is all planes of
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the flock share a common line, then the resulting generalised quadrangle is isomorphic to H(3, q 2 ). There exist nonlinear flocks, and so nonclassical flock quadrangles, and these have been classified for all q 37, q = 32 (see [6], [11], [2]). For instance, if q is a prime power congruent to 2 modulo 3, there exist non-linear flock quadrangles known as the Fisher–Thas– Walker–Kantor generalised quadrangles. We will use a construction given in [1] for the dual of this generalised quadrangle, which we will denote by FTWK(q). In the next section, we provide some theory and background material on the Hermitian surface H(3, q 2 ). In particular, we revise the representation of the Fisher–Thas–Walker–Kantor quadrangles on the Hermitian surface, that was given in [1]. It is this representation which allows us to relate our new hemisystem of FTWK(5) to the set of lines of H(3, 52 ) dual to the 3 · A7 -hemisystem. In Sects. 4 and 5, we give constructions of the two hemisystems of this paper (in the generalised quadrangles FTWK(5) and H(3, 52 ) respectively). The notation we use in this paper for projective and polar spaces is standard, but the reader can find more details on the theory of projective and polar spaces in a text such as [10]. We will often think of a geometric object (such as a line) as a set of points for expedient notation; for example, we will sometimes write ∩ O for the point of the ovoid O which lies on the line .
2 The Fisher–Thas–Walker–Kantor quadrangles The Hermitian variety H(3, q 2 ) is a generalised quadrangle defined by a Hermitian form, q q q q such as X, Y = X 1 Y1 + X 2 Y2 + X 3 Y3 + X 4 Y4 . This geometry is rich in that there are subgeometries which are equivalent to projective spaces over GF(q), rather than GF(q 2 ). This generalised quadrangle has order (q 2 , q), and so has (q 3 + 1)(q 2 + 1) points and (q 3 + 1)(q + 1) lines that are called generators. Moreover, the dual of this generalised quadrangle is isomorphic to the geometry of points and lines of the elliptic quadric Q− (5, q). There are three different ways a line of PG(3, q 2 ) can intersect H(3, q 2 ): in a generator of H(3, q 2 ), in one point or in a Baer subline; that is, a subset of q + 1 points in a PG(1, q 2 ) forming an isomorphic copy of the projective line PG(1, q). A Baer subgenerator of H(3, q 2 ) is a Baer subline of a generator of H(3, q 2 ). The Hermitian form given above also defines a polarity of the ambient projective space, which we denote by u (n.b., we use the customary symbol ⊥ for a orthogonal polarity in a later section). The generators incident with a point X of H(3, q 2 ) are contained in the plane X u . Moreover, the planes of PG(3, q 2 ) come in just two types with respect to u; (i) a degenerate plane of q + 1 generators on a point (also known as a Baer subpencil of lines), or (ii) a nondegenerate plane consisting only of a Hermitian curve H(2, q 2 ). From [1], we have a construction of the dual of the Fisher–Thas–Walker–Kantor generalised quadrangles as substructures of H(3, q 2 ). We begin with a fixed Hermitian curve O and a point P of O, and we consider a distance metric on points, lines and Baer subgenerators of H(3, q 2 ) as follows: Distance from P
Substructures
1 2 3 4 5 6
Generators on P Baer subgenerators on P Points in P u \{P} Baer subgenerators not on P, with a point in P u Generators of H(3, q 2 ) not in P u ; the points not in O ∪ P u Baer subgenerators with no point in P u ; the points of O\{P}
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This distance metric actually arises from a realisation of the Split Cayley hexagon as a substructure of H(3, q 2 ) and more information can be found in [1]. Below we define an important notion in this paper: Definition 2.1 (Ischia Set) Fix a Hermitian curve O and a point P of O. Let b be a Baer subgenerator of H(3, q 2 ) at distance 6 from P containing a point of O. For each X ∈ b, not in O, let X be the Baer subline X u ∩ O. There is a unique point PX of P u in the Baer subplane spanned by b and X . Then the Ischia set I (b) with respect to O and P determined by b is the set of q lines X PX , together with the generator spanned by b. One can show (see [1]) that the Ischia set determined by b meets P u in a Baer subconic. In [1], an equivalence relation on Baer subgenerators was constructed which yields q + 1 different equivalence classes. These equivalence classes can also be realised as the orbits of the subgroup SU(3, q 2 ) of the stabiliser of the ovoid O. With this setup, there is then a nice description of the Fisher–Thas–Walker–Kantor generalised quadrangles in H(3, q 2 ). Definition 2.2 (FTWK Generalised Quadrangle) Let q be a prime power congruent to 2 modulo 3. Fix a Hermitian curve O and a point P of O, and let be an equivalence class of Baer subgenerators at distance 6 to P, each meeting O. Then the following defines the Fisher–Thas–Walker–Kantor generalised quadrangle FTWK(q): Points: Lines: Incidence:
The generators of H(3, q 2 ). (a) Points of O ∪ P u ; (b) Ischia sets (with respect to O and P) arising from . Natural incidence or inclusion.
In this model, we can easily describe the defining properties of a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle FTWK(q), q odd. Lemma 2.1 Let q be an odd prime power congruent to 2 modulo 3, let O be a Hermitian curve of H(3, q 2 ), let P be a point of O and let be an equivalence class of Baer subgenerators at distance 6 to P meeting O. A set of generators L of H(3, q 2 ) defines a hemisystem of FTWK(q) if and only if the following sets of lines contain precisely (q + 1)/2 elements of L: (i) for a point X of O, the set of lines on X ; (ii) for a point X of P u , the set of lines on X ; (iii) Ischia sets arising from .
3 Commuting polarities The notion of a commuting pair of polarities, one Hermitian and the other orthogonal, plays a central role in the constructions given in [5]. It so happens that this phenomenon appears in the work of this paper, and we give a brief overview of the necessary background on this subject below. Let q be an odd prime power, let C be a nonsingular conic of the plane π = PG(2, q 2 ) and let be a line of π. From C and , one can produce more conics of π as follows. Since C is a nonsingular conic, there is an irreducible homogeneous polynomial of degree 2 associated with C . Likewise, the square of the homogeneous equation of is also such a polynomial, and a linear combination of the two induces a conic C + t2 of π, where t ∈ GF(q 2 ). By a result of Segre [12], if a conic commutes with the unitary polarity
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defining a Hermitian curve H, then the intersection of H with is a Baer subconic of . Recently, Donati and Durante [8] have proved the converse, that if a Hermitian curve and a conic meet in a Baer subconic, then the two polarities commute.
4 An explicit construction of a hemisystem of a Fisher–Thas–Walker–Kantor quadrangle Without loss of generality, we may suppose that the unitary polarity u of PG(3, q 2 ) defining q q q q H(3, q 2 ) arises from the form X, Y = X 1 Y1 + X 2 Y2 + X 3 Y3 + X 4 Y4 . Let π be the plane X 4 = 0, that is, the polar plane of the point (0, 0, 0, 1) with respect to u; note that it is a plane which meets H(3, q 2 ) in a Hermitian curve O. From a certain partition of the points of O, we construct a set of lines as follows. Construction 4.1 Let q be an odd prime power, let O be a Hermitian curve of H(3, q 2 ), let P be a point of O, and let L1/2 be a set of half the generators on P. Suppose we have a partition of O into {P}, S1 and S2 such that every Baer secant to O on P meets S1 in (q −1)/2 points, and meets S2 in (q + 1)/2 points. Let L be the set of all generators satisfying the following properties. If ∩ O ∈ S1 ,
then is concurrent with some element of L1/2 .
If ∩ O ∈ S2 , then is not concurrent with any element of L1/2 . Then L ∪ L1/2 consists of half the generators of H(3, q 2 ) satisfying conditions (i) and (ii) of Lemma 2.1. It remains to give an explicit description of one of the parts S1 or S2 , such that the third condition of Lemma 2.1 is also satisfied. Let G = PGU(4, q2 ) and consider the kernel of the action G (π ) of G on the points of π. Note that G (π ) consists (projectively) of matrices of the form ⎞ ⎛ 1000 ⎜0 1 0 0 ⎟ ⎟ Mλ := ⎜ ⎝0 0 1 0 ⎠ 000λ where λq+1 = 1. By taking those Mλ which satisfy λ(q+1)/2 = 1 we obtain a cyclic regular subgroup of G (π ) of order (q + 1)/2. So there is a natural partition of the q + 1 lines on P into two halves; let L1/2 be one of these halves. Let ⊥ be an orthogonal polarity commuting with u, say the one induced by the form (X 1 , X 2 , X 3 , X 4 ) → X 12 + X 22 + X 32 + X 42 . Let C be the conic of totally singular points of π with respect to ⊥ and note that by a result of Segre [12], C intersects the unital O in a subconic of points which have coordinates in the ground field GF(q). We suppose that P = (1, i, 0, 0) is one of the points of this subconic, where i is some element of GF(q) satisfying i 2 + 1 = 0; which implies that we assume q ≡ 1 (mod 4). We will now split the points lying on secants P, Q to the subconic (i.e., Q ∈ C ∩ O) into two halves. Lemma 4.1 Let P be the tangent line P u ∩ π at P. The conics C + t2P where t q + t = 0, commute with u and cover precisely the points lying on lines on P secant to the subconic C ∩ O.
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Proof For the following, we restrict our attention to the plane π : X 4 = 0 and use homogeneous triples rather than four coordinates. Let P = (1, i, 0) where i 2 + 1 = 0. and let C be a conic defined by the following matrix over GF(q 2 ): ⎛ ⎞ α β −i ⎝ β α + 2iβ ⎠ . −i 1 Then it is a simple exercise to show that the polarity defining C commutes with the unitary polarity u if and only if α q+1 + β q+1 − q+1 = 1, (α + βi)q+1 = 1 and (α + βi) q = −. Moreover, every conic on P commuting with u is of this form. Let Ct denote one of the q + 1 conics defined by C + t2P where t q + t = 0. Since C : X 12 + X 22 + X 32 = 0 and P : X 1 + i X 2 = 0 it follows that Ct : (1 + t)X 12 + (1 − t)X 22 + X 32 + 2it X 1 X 2 = 0. Hence Ct commutes with the unitary polarity of O and so Ct ∩ O is a Baer subconic on P. Let P be the Baer subpencil with vertex P containing the lines P X with X ∈ C ∩ O and the line P . We just need to show that every line k of P intersects every conic Ct in a unique point Rt,k (other than P) and that the point Rt,k is in O. Now a generic point X on (C ∩ O)\{P} has the following coordinates X = (1−s 2 , −i(1+ s 2 ), 2s), where s ∈ GF(q) and hence the q lines on P , different from P have the following form: k : X 1 = (1 − s 2 ) + k;
X 2 = −i(1 + s 2 ) + ki;
X 3 = 2s.
In order to obtain the point (different from P) in k ∩ Ct , we have to solve the following equation: (1 + t)(1 − s 2 + k)2 − (1 − t)(−1 − s 2 + k)2 + 4s 2 − 2t (1 − s 2 + k)(−1 − s 2 + k) = 0, that is, t + k = 0. Then k = −t and Rt,k = (1 − s 2 − t, −i(1 + s 2 + t), 2s) and it is easy to see that Rt,k ∈ O. q We can partition these points further into two parts. Since t + t = 0, and t = 0 for points not equal to P, we have two values for t (q−1)/2 : i and −i. Let Ti be the set of points of O covered by the conics C + t2P for t (q−1)/2 = i, and similarly, define T−i to be those covered by the other type of conics (t (q−1)/2 = −i). Thus we get our desired partition (Table 1 and Fig. 1). It turns out by computer that if q = 5, the set of generators we obtain when we apply Construction 4.1 with the above partition, satisfies condition (iii) of Lemma 2.1. Thus we obtain a hemisystem of FTWK(5) and the main theorem of this paper (see Sect. 1). Remark 4.1 (i) It is possible that this construction does not yield hemisystems of the Fisher– Thas–Walker–Kantor quadrangles for larger values of q, and we will see in the next section, that there is evidence that our example is sporadic. One of the main problems in trying to construct an infinite family of hemisystems of FTWK(q) is that condition (iii) of Lemma 2.1 is difficult (at this stage) to verify. (ii) The stabiliser of the ovoid O in PU(4, q 2 ) is GU(3, q 2 ).φ where φ is the Frobenius automorphism of GF(q 2 ) (note that the kernel of the action on the ovoid is cyclic of Table 1 Partition of O yielding a hemisystem of FTWK(5) S1 S2
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On secants to the subconic
Off secants to the subconic
T−i Ti ∪ (C ∩ O)\{P}
Internal points External points
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Fig. 1 Squares denote points of C ∩ O, diamonds (resp. circles) denote internal (resp. external) points, black points denote elements of S2 , and white points denote elements of S1
order q + 1). If we then look at the stabiliser of C inside this group, we obtain a subgroup isomorphic to PGL(2, q) × (Cq+1 : φ), and the subgroup thereof which fixes P is simply the one-dimensional affine subfield group AGL(1, q) × (Cq+1 : φ) . Thus the stabiliser of our hemisystem is isomorphic to AGL(1, q) × (C(q+1)/2 : C2 ) ∼ = AGL(1, 5) × S3 . 5 The 3 · A7 -hemisystem of Q− (5, 5) For q = 5, the alternating group A7 appears curiously as a maximal subgroup of PSU(3, q 2 ) and Dye has given a beautiful account of this phenomenon in his paper [9] of 1999, although, this representation of A7 was known at least as far back as Mitchell in 1911 (for more information, see [9] and the references within). Dye constructs a partition of the classical unital H(2, 52 ) into 21 Clebsch hexagons; which are in fact Baer subconics of H(2, 52 ), and hence the full conics defined by these subconics commute with H(2, 52 ) (see Sect. 3). These Clebsch hexagons can be chosen such that one of them is the distinguished subconic defined by the conic C : X 12 + X 22 + X 32 = 0, ten of them partition the internal points to this conic and the remaining ten partition the external points to this conic. We show now that there is a strong relationship between the hemisystem of FTWK(5) constructed in the previous section and the 3· A7 -hemisystem of Q− (5, 5). Recall from Sect. 4 the construction of the partition of the unital into {P}, S1 and S2 . Within S2 , we have a special set E of 26 points which we describe as follows. There are q 2 (q + 1) = 150 conics on P which commute with the Hermitian polarity. Of these, only q + 1 = 6 have their coefficients in GF(q) and they are precisely those which have at least two points of the subconic (see Sect. 3). The 5 conics not equal to our original subconic cover 20 points of the unital, external
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Table 2 Partition of O yielding the 3 · A7 -hemisystem S1 S2
On secants to the subconic
Off secants to the subconic
Ti T−i ∪ (C ∩ O)\{P}
All external points except E Internal points together with E\O
to the conic C . So we have a distinguished set E of 26 points of the unital, containing the subconic C ∩ O. We can use the partition of the unital to construct the 3 · A7 -hemisystem of Q− (5, 5) by considering the following partition of O and the method described in Sect. 4 for generating half the generators of H(3, q 2 ). Theorem 5.1 The partition given in Table 2 defines a hemisystem of lines of H(3, 52 ) via Construction 4.1, equivalent to the 3 · A7 -hemisystem. Proof By computer.
We make the following observation which allows us to distinguish the partial quadrangle arising from our hemisystem of FTWK(5) from the 3 · A7 -hemisystem. An embedding of a point-line incidence structure S into a projective space PG(n, q) is an injective map on the point set of S into the point set of PG(n, q) such that the points on a line of S are mapped to the points of a common line of PG(n, q). Lemma 5.1 Let P be a partial quadrangle PQ((q − 1)/2, q 2 , (q − 1)2 /2) where q is an odd prime power no smaller than 5. If there exists an embedding of P into Q− (5, q), then the image of this embedding is a hemisystem of Q− (5, q). Proof Every line of P has at least three points and so the image of a line of P must span a totally singular line of Q− (5, q). We must check that two different lines of P are not mapped onto the same line of Q− (5, q). Suppose by contradiction that we have two distinct lines and of P which are both mapped to the same line m of Q− (5, q). Now there exists a pair of noncollinear points (X, X ) such that X ∈ and X ∈ . Since the µ value of P is nonzero, there exists a point Y collinear to both X and X . However, when one considers the images of X , X and Y in Q− (5, q), there appears a triangle in Q− (5, q) as X and X are mapped to collinear points; a contradiction since Q− (5, q) is a generalised quadrangle. Therefore, two different lines of P are not mapped onto the same line of Q− (5, q). Since the number of lines of P is equal to the number of lines of Q− (5, q) (namely, (q 3 + 1)(q 2 + 1)), the result follows. Remark 5.1 (i) Suppose we have a partition of the unital into {P}, S1 and S2 , and suppose that the set of generators H arising from this partition (via Construction 4.1) is a hemisystem of lines of H(3, q 2 ). Since every point of H(3, q 2 ) is incident with (q + 1)/2 elements of H, it follows that every Baer subline of O must meet {P} ∪ S1 in an even number of points. Our partial quadrangle constructed in Sect. 2.1 is not a hemisystem of lines of H(3, 52 ) since there are 160 Baer sublines in the plane of O which contain an odd number of points in the part S1 of the partition. Therefore, by Lemma 5.1, the partial quadrangle arising from our hemisystem of FTWK(5) cannot be embedded in Q− (5, 5). (ii) As mentioned above, the connection to the 3 · A7 -hemisystem suggests that our hemisystem of FTWK(q) is a sporadic example. However, the construction we used in Sect. 2.1 has natural ingredients which exist for every q ≡ 5 (mod 12). It would be very interesting if one could prove that our construction yields a hemisystem of FTWK(17) or for higher values of q. As for generalising the 3 · A7 -hemisystem, when using the construction outlined
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by Table 2 for q ∈ {9, 13, 17, 25}, it turns out by computer that we do not get a hemisystem of lines of H(3, q 2 ). (iii) The stabiliser in (3· A7 ).2 of a point P in the ovoid O fixed by (3· A7 ).2, is the stabiliser of our hemisystem of FTWK(5), which we remarked before is isomorphic to AGL(1, 5)× S3 . The partial quadrangle and strongly regular graph which arise also have this group as their automorphism groups. (iv) From our hemisystem H of FTWK(5), we obtain a new antipodal cometric association scheme with 4-classes defined by the following relations on points of FTWK(5): R1 R2 R3 R4
both in the same half (H or its complement) and collinear; both in the same half and not collinear; in different halves and collinear; in different halves and not collinear.
We would like to thank Bill Martin for pointing out this connection. Acknowledgments This work was supported by the GOA-grant “Incidence Geometry” at Ghent University. The first author acknowledges the support of a Marie Curie Incoming International Fellowship within the 6th European Community Framework Programme (contract number: MIIF1-CT-2006-040360), and the third author was supported by a travel fellowship from the School of Sciences and Technology – University of Naples “Federico II”.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Bamberg J., Durante N.: On the Fisher-Thas-Walker-Kantor generalised quadrangles (submitted). Betten A.: http://www.math.colostate.edu/~betten/blt.html. Cameron P.J.: Partial quadrangles. Quart. J. Math. Oxford Ser. II 26, 61–73. Cameron P.J., Delsarte P., Goethals J.-M.: Hemisystems, orthogonal configurations, and dissipative conference matrices. Philips J. Res. 34(3–4), 147–162 (1979). Cossidente A., Penttila T.: Hemisystems on the Hermitian surface. J. London Math. Soc. II 72(3), 731– 741 (2005). De Clerck F., Van Maldeghem H.: Some classes of rank 2 geometries. In: Buekenhout F. (ed.) Handbook of Incidence Geometry, pp. 433–475. North-Holland, Amsterdam (1995). De Clerck F., Gevaert H., Thas J.A.: Flocks of a quadratic cone in PG(3,q), q ≤ 8. Geom. Dedicata 26, 215–230 (1988). Donati G., Durante N.: On the intersection of a Hermitian curve with a conic (submitted). Dye R.H.: Some geometry of A7 and PSU3 (52 ). J. Geom. 65(1–2), 77–90 (1999). Hirschfeld J.W.P., Thas J.A.: General Galois geometries. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1991). Law M., Penttila T.: Classification of flocks of the quadratic cone over fields of order at most 29. Adv. Geom., (suppl.), S232–S244 (2003); Special issue dedicated to Adriano Barlotti. Segre B.: Forme e geometrie hermitiane, con particolare riguardo al caso finito. Ann. Mat. Pura Appl. 70(4), 1–201 (1965). Thas J.A.: Generalized quadrangles and flocks of cones. European J. Combin. 8(4), 441–452 (1987).
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