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Some generalization of Desargues and Veronese configurations. (English) Zbl 1164.51302

The classical Desargues configuration \(10_3\) consists of three lines of size 3 through a point \(p\), three Veblen configurations inscribed into every pair of the given lines, and an axis which joins corresponding points of intersection. Next to the classical Desargues configuration, also the combinatorial Veronese space \(\mathbf{V}_3(3)\) satisfies these properties. Then the classical Desargues configuration and the combinatorial Veronese space \(\mathbf{V}_3(3)\) are the only two configurations having these properties.
The authors generalize this configuration. Given a set of points, and a given point \(p\), define a set \(\mathcal{L}_p\) of triples of points, called lines, all containing the point \(p\). What configurations can occur if every pair of these lines yields a Veblen figure?
This leads to a study of partial triple systems, more precisely \((v_r,b_3)\) configurations, i.e., configurations of \(v\) points such that every point lies on \(r\) lines, together with \(b\) lines consisting of three points. They study how these configurations look like, and study automorphism groups of these configurations. In particular, some \(\begin{pmatrix} n+2\\2\end{pmatrix}_n,\begin{pmatrix} n+2\\3\end{pmatrix}_3\) configurations are constructed.
Reviewer: Leo Storme (Gent)

MSC:

51E14 Finite partial geometries (general), nets, partial spreads
51E30 Other finite incidence structures (geometric aspects)
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