Geodesic
Hermitian Configurations in Odd-Dimensional Projective Geometries
Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 500-512

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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, q 2) is a disjoint union of (q 2n+l – l)/(q – 1) (q 2n+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, q 2)|. 
Kestenband, Barbu C. Hermitian Configurations in Odd-Dimensional Projective Geometries. Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 500-512. doi: 10.4153/CJM-1981-043-7
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[3] 3. Kestenband, B. C., Projective geometries that are disjoint unions of caps, Can. J. Math. (to appear). Google Scholar

[4] 4. Singer, James, A theorem infinite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385. Google Scholar

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