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Bundles of conics derived from planar projective incidence groups. (English) Zbl 0882.51001

The authors study Desarguesian projective planes \(\mathcal P\) that are endowed with a linear incidence group, i.e. the set of points of \(\mathcal P\) forms a group \(G\) and for \(a \in G\) the mappings \(x \mapsto ax, x \mapsto xa: G \to G\) are collineations of \(\mathcal P\). The results are mainly based on the following representation theorem: there exists a commutative cubic field extension (\(K(\varepsilon),K)\) such that the points of \(\mathcal P\) are just the one-dimensional \(K\)-subspaces of \(K(\varepsilon)\) and the lines are the two-dimensional subspaces. Moreover, the group \(G\) is canonically isomorphic to the quotient of the multiplicative groups of \(K(\varepsilon)\) and \(K\).
The main part of the paper is devoted to the investigation of the mapping \(\rho: x \mapsto x^r: G \to G\) for rational numbers \(r\). It is proved that \(\rho(L)\) is always an algebraic curve. It is a conic for any line \(L\), if \(r = -1,2,2^{-1}\) and a line when the characteristic of \(K\) divides the exponent \(r\). Finally, if \(\rho\) is injective, the family \(\mathcal L\) of all images \(\rho(L)\) of lines \(L\) form the set of lines of a new projective plane \({\mathcal P}_\rho = (G,\mathcal L)\).

MSC:

51A35 Non-Desarguesian affine and projective planes
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