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Partitioning Projective Geometries into Caps

Published online by Cambridge University Press:  20 November 2018

Gary L. Ebert
Gary L. Ebert*
Affiliation:
University of Delaware, Newark, Delaware
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In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q2) can be partitioned into (q2n++ 1 + 1)/(q + l)-caps. Moreover, these caps were shown to constitute the “large points” of a PG(2n, q) in a natural way. In [3] a similar argument was used to show that once two disjoint (n – l)-subspaces are removed from PG(2n, q2), the remaining points can be partitioned into (q2n – 1)/(q2 – l)-caps.

The purpose of this paper is to give a short proof of the results found in [2], and then use the technique developed to partition PG(2n, q) into (qn + l)-caps for n even and q any prime-power. Moreover, these caps can be treated in a natural way as the “large points” of a PG(n – 1, q).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 6 , 01 December 1985 , pp. 1163 - 1175
Copyright
Copyright © Canadian Mathematical Society 1985

References

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2. Kestenband, B. C., Projective geometries that are disjoint unions of caps, Can. J. Math. 32 (1980), 1299–1305.Google Scholar
3. Kestenband, B. C., Hermitian configurations in odd-dimensional projective geometries, Can. J. Math. 33 (1981), 500–512.Google Scholar
4. Qvist, B., Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn. 734 (1952), 1–27.Google Scholar