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Hermitian Configurations in Odd-Dimensional Projective Geometries

Published online by Cambridge University Press:  20 November 2018

Barbu C. Kestenband
Barbu C. Kestenband*
Affiliation:
New York Institute of Technology, Old Westbury, New York
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A t-cap in a geometry is a set of t points no three of which are collinear. A (t, k)-cap is a set of t points, no k + 1 of which are collinear.

It has been shown in [3] that any Desarguesian PG(2n, q2) is a disjoint union of (q2n+l – l)/(q – 1) (q2n+l – l)/(q + l)-caps. These caps were obtained as intersections of 2n Hermitian Varieties of a certain kind; the intersection of 2n + 1 such varieties was empty. Furthermore, the caps in question constituted the ‘large points” of a PG(2n, q), with the incidence relation defined in a natural way.

It seemed at the time that nothing similar could be said about odd-dimensional projective geometries, if only because |PG(2n – 1, q)| ∤ |PG(2n – l, q2)|.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 500 - 512
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Birkhoff, G. and MacLane, S., A survey of modern algebra (MacMillan, Third Ed., 1966).Google Scholar
2. Bose, R. C. and Chakravarti, I. M., Hermitian varieties in a finite projective space PG(N, q*), Can. J. Math. 18 (1966), 1161–1182.Google Scholar
3. Kestenband, B. C., Projective geometries that are disjoint unions of caps, Can. J. Math. (to appear).Google Scholar
4. Singer, James, A theorem infinite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.Google Scholar