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On the size of maximal caps in \(Q^+(5,q)\). (English) Zbl 1043.51008

The paper of Metsch provides an interesting contribution to the study of caps in finite quadrics.
A cap \({\mathcal C}\) of the hyperbolic quadric \(Q^+(5, q)\) is a set of points of \(Q^+(5, q)\) that does not contain three collinear points. It is called maximal if it is not contained in a larger cap.
For \(q\) odd it is easily seen that \(| {\mathcal C}| \leq q^3+q^2+q+1\). For \(q\) odd D. G. Glynn [Geom. Dedicata 26, No. 3, 273–280 (1988; Zbl 0645.51012)] constructed caps of size \(q^3+q^2+q+1\) and L. Storme [J. Comb. Theory, Ser. A 87, No. 2, 357–378 (1999; Zbl 0947.51009)] characterized caps of size \(q^3+q^2+q+1\) as intersections of \(Q^+(5, q)\) with another quadric, if \(q \geq 3139\).
Metsch deals with the problem of the maximal cardinality of caps of \(Q^+(5, q)\), \(q\) odd, with \(| {\mathcal Q}| < q^3+q^2+q+1\). His main result is as follows:
Theorem. Let \({\mathcal C}\) be a maximal cap of \(Q^+(5, q)\), \(q\) odd, with \(| {\mathcal Q}| < q^3+q^2+q+1\). If \(q \geq 4364\), then \(| {\mathcal Q}| \leq q^3+q^2+2\).
Note that Storme constructed a maximal cap of size \(q^3+q^2+1\). It is not clear, whether maximal caps of size \(q^3+q^2+2\) do exist. Metsch obtained the following information about such an hypothetical cap:
Theorem. Suppose that there exists a maximal cap \({\mathcal C}\) of \(Q = Q^+(5, q)\), \(q\) odd, of size \(q^3+q^2+2\). If \(q \geq 4364\), then one of the following cases occurs:
(i) There exists a line \(h\) of \(Q\) with \(| h \cap {\mathcal C}| = 2\). The two planes of \(Q\) through \(h\) meet \({\mathcal C}\) in the two points of \(h \cap {\mathcal C}\). Every plane of \(Q\) intersecting \(h\) in a point \(p \notin {\mathcal C}\) intersects \({\mathcal C}\) in a \(q\)-arc that can be extended to a conic by adjoining \(p\). All other planes of \(Q\) intersect \({\mathcal C}\) in a conic.
(ii) There exists a point \(p \in {\mathcal C}\) such that all planes of \(Q\) through \(p\) meet \({\mathcal C}\) in two points whereas all other planes of \(Q\) meet \({\mathcal C}\) in a conic.

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs
51E20 Combinatorial structures in finite projective spaces
51E12 Generalized quadrangles and generalized polygons in finite geometry
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