On the size of maximal caps in Q + (5, q)

Klaus Metsch
2003 Advances in Geometry  
Let q be an odd prime power. A cap of the hyperbolic quadric Q þ ð5; qÞ is a set of points of Q þ ð5; qÞ that does not contain three collinear points. It is called maximal, if it is not contained in a larger cap. It is easy to see that a cap has size at most q 3 þ q 2 þ q þ 1 with equality if and only if it meets every plane of Q þ ð5; qÞ in a conic. Caps of this size do exist. The largest known maximal cap with less than q 3 þ q 2 þ q þ 1 points has size q 3 þ q 2 þ 1. In this paper it is
more » ... is paper it is shown that for large odd q all caps with more than q 3 þ q 2 þ 2 points are contained in a cap of size q 3 þ q 2 þ q þ 1. Also, some structural information is given on a hypothetical maximal cap of size q 3 þ q 2 þ 2. This result is an analogue of an extension result in the case that q is even. In contrast to the even case, the odd case heavily relies on algebraic arguments. On the size of maximal caps in Q þ ð5; qÞ S187
doi:10.1515/advg.2003.2003.s1.186 fatcat:lokvgffatbdbdbfid7dniegwpq