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Some remarks on flocks. (English) Zbl 1056.51002

A partial BLT-set of a generalized quadrangle \(Q\) of order \((s,t)\), \(s,t > 1\) is a set \(B\) of at least three points such that every point of \(Q\) is collinear with at most two points of \(B\). A partial BLT-set has size at most \(s+1\), in which case it is called a BLT-set. Consider in \(\text{PG}(2n-1,q)\) a cone \({\mathcal K}\) with vertex \(v\) and base a parabolic quadric \(Q(2n-2,q)\) in a hyperplane not containing \(v\). A partial flock of \({\mathcal K}\) is a set \(\{ \pi_1,\ldots,\pi_k \}\) of \(k\) distinct hyperplanes such that \(\pi_i \cap \pi_j \cap \mathcal{K}\) is a nonsingular elliptic quadric of \(\pi_i \cap \pi_j\) for all \(i,j \in \{ 1,\ldots,k \}\) with \(i \not= j\).
In the paper, the authors give new proofs for results of L. Bader, G. Lunardon and J. A. Thas [Forum Math. 2, 163–174 (1990; Zbl 0692.51006)] which relate flocks of the quadratic cone in \(\text{ PG}(3,q)\), \(q\) odd, and BLT-sets of \(Q(4,q)\). They also show that there exists a unique BLT-set in the classical generalized quadrangle \(H(3,9)\). For \(q\) congruent to \(1\) or \(3\) modulo 8 and \(n\) even, partial flocks of size \(\frac{n(q+1)}{2}-1\) of the cone in \(\text{PG}(2n-1,q)\) are given.
In [J. Algebr. Comb. 6, 377–392 (1997; Zbl 0897.51007)], C. M. O’Keefe and J. A. Thas gave a partial flock of size 6 in PG(5,3). The authors of the present paper show that this partial ovoid belongs to an infinite class: for any prime \(p\) congruent to 1 or 3 modulo 8 and every \(n \geq 1\) they construct a partial flock of size \(2pn\) in \(\text{ PG}(2pn-1,p)\).

MSC:

51E12 Generalized quadrangles and generalized polygons in finite geometry
51A40 Translation planes and spreads in linear incidence geometry
51A50 Polar geometry, symplectic spaces, orthogonal spaces
51E20 Combinatorial structures in finite projective spaces
05B25 Combinatorial aspects of finite geometries

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References:

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