Partitioning Projective Geometries into Caps

Gary L. Ebert

doi:10.4153/CJM-1985-063-1

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In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q 2) can be partitioned into (q 2n++ 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, q 2), the remaining points can be partitioned into (q 2n – 1)/(q 2 – 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).

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References

1. Barlotti, A., Un'estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. 10 (1955), 498–506.Google Scholar

2. Kestenband, B. C., Projective geometries that are disjoint unions of caps, Can. J. Math. 32 (1980), 1299–1305.Google Scholar

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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

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