## 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

### Keywords:

conics; projective plane; cyclic difference set
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