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Projective bundles. (English) Zbl 0803.51009

It is well-known that there exist collections of \(q^ 2+q+1\) conics of \(\text{PG} (2,q)\) which mutually intersect in a unique point and thus form another projective plane of order \(q\) on the same point set; the authors call such a collection of conics a projective bundle. Motivated by the question, whether one may partition the \(q^ 5 - q^ 2\) conics of \(\text{PG} (2,q)\) into \(q^ 2(q-1)\) projective bundles, they construct collections of \(q^ 2(q - 1)/2\) disjoint projective bundles for any odd prime power \(q\); for even \(q\), they can only obtain \(q-1\) disjoint bundles. In the cases \(q=3\) or 4, better results (but no complete partitioning) are obtained. The authors also give the following interesting result: If \(D\) is a cyclic difference set for \(\text{PG} (2,q)\) and if \(r\) is relatively prime to \(q^ 2 + q + 1\), then \(Dr^{- 1}\) is a curve of degree \(r\).

MSC:

51E15 Finite affine and projective planes (geometric aspects)
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B25 Combinatorial aspects of finite geometries
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