Small point sets that meet all generators of \(Q(2n,p)\), \(p>3\) prime. (English) Zbl 1052.51005
Let \(Q(2n,q)\) be the non-singular parabolic quadric in \(PG(2n,q)\). An ovoid of \(Q(2n,q)\) is a set of points of \(Q(2n,q)\) intersecting every generator of \(Q(2n,q)\) in exactly one point. A blocking set of \(Q(2n,q)\) is a set of points of \(Q(2n,q)\) intersecting every generator of \(Q(2n,q)\) in at least one point.
It has been shown by Ball, Govaerts and Storme that every ovoid of \(Q(4,q)\), \(q\) prime, is a 3-dimensional elliptic quadric. A result of Christine M. O’Keefe and J. A. Thas [Eur. J. Comb. 16, No. 1, 87–92 (1995; Zbl 0819.51005)] then implies that \(Q(6,q)\), \(q>3\) prime, does not contain ovoids.
The authors prove in this article that the smallest blocking sets of \(Q(6,q)\), \(q>3\) prime, are truncated cones \(PQ^-(3,q)\setminus \{P\}\), with \(P\) a point of \(Q(6,q)\) and with \(Q^-(3,q)\) a 3-dimensional elliptic quadric in the tangent hyperplane to \(Q(6,q)\) in \(P\).
It then follows from results of De Beule and Storme that the smallest blocking sets of \(Q(2n,q)\), \(q>3\) prime, \(n>3\), are equal to truncated cones \(S_{n-3}Q^-(3,q)\), with \(S_{n-3}\) an \((n-3)\)-dimensional space contained in \(Q(2n,q)\), and with \(Q^-(3,q)\) a 3-dimensional elliptic quadric contained in the polar space of \(Q(2n,q)\) in \(S_{n-3}\).
It has been shown by Ball, Govaerts and Storme that every ovoid of \(Q(4,q)\), \(q\) prime, is a 3-dimensional elliptic quadric. A result of Christine M. O’Keefe and J. A. Thas [Eur. J. Comb. 16, No. 1, 87–92 (1995; Zbl 0819.51005)] then implies that \(Q(6,q)\), \(q>3\) prime, does not contain ovoids.
The authors prove in this article that the smallest blocking sets of \(Q(6,q)\), \(q>3\) prime, are truncated cones \(PQ^-(3,q)\setminus \{P\}\), with \(P\) a point of \(Q(6,q)\) and with \(Q^-(3,q)\) a 3-dimensional elliptic quadric in the tangent hyperplane to \(Q(6,q)\) in \(P\).
It then follows from results of De Beule and Storme that the smallest blocking sets of \(Q(2n,q)\), \(q>3\) prime, \(n>3\), are equal to truncated cones \(S_{n-3}Q^-(3,q)\), with \(S_{n-3}\) an \((n-3)\)-dimensional space contained in \(Q(2n,q)\), and with \(Q^-(3,q)\) a 3-dimensional elliptic quadric contained in the polar space of \(Q(2n,q)\) in \(S_{n-3}\).
Reviewer: Leo Storme (Gent)
MSC:
| 51E21 | Blocking sets, ovals, \(k\)-arcs |
| 05B25 | Combinatorial aspects of finite geometries |
| 51E20 | Combinatorial structures in finite projective spaces |
Citations:
Zbl 0819.51005References:
| [1] | S. Ball, On ovoids of O(5,\(q\); S. Ball, On ovoids of O(5,\(q\) |
| [2] | S. Ball, P. Govaerts, L. Storme, On ovoids of \(QqQqq\); S. Ball, P. Govaerts, L. Storme, On ovoids of \(QqQqq\) |
| [3] | J. De Beule, K. Metsch, Small point sets that meet all generators of \(Hnq^2\); J. De Beule, K. Metsch, Small point sets that meet all generators of \(Hnq^2\) · Zbl 1072.51004 |
| [4] | J. De Beule, L. Storme, The smallest minimal blocking sets of \(Qnqq\); J. De Beule, L. Storme, The smallest minimal blocking sets of \(Qnqq\) · Zbl 1075.51001 |
| [5] | Gunawardena, A.; Moorhouse, G. E., The non-existence of ovoids in \(O_9(q)\), European J. Combin., 18, 2, 171-173 (1997) · Zbl 0872.51005 |
| [6] | Metsch, K., The sets closest to ovoids in \(Q^−(2n+1,q)\), Bull. Belg. Math. Soc. Simon Stevin, 5, 2-3, 389-392 (1998), Finite geometry and combinatorics, Deinze, 1997 · Zbl 0927.51016 |
| [7] | Metsch, K., On blocking sets of quadrics, J. Geom., 67, 1-2, 188-207 (2000), Second Pythagorean Conference (Pythagoreion, 1999) · Zbl 0970.51010 |
| [8] | K. Metsch, Small point sets that meet all generators of \(Wnq\); K. Metsch, Small point sets that meet all generators of \(Wnq\) · Zbl 1052.51007 |
| [9] | O’Keefe, Ch. M.; Thas, J. A., Ovoids of the quadric \(Q(2n,q)\), European J. Combin., 16, 1, 87-92 (1995) · Zbl 0819.51005 |
| [10] | Thas, J. A., Ovoids and spreads of finite classical polar spaces, Geom. Dedicata, 10, 1-4, 135-143 (1981) · Zbl 0458.51010 |
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