One-point extensions in \(n_3\) configurations (Q2827781)

An \textit{\(n_3\) configuration} is a projective configuration with \(n\) points and \(n\) lines such that every line is incident on \(3\) points and every point is incident on \(3\) lines. Given an \(n_3\) configuration, a \(1\)-point extension is a technique that constructs an \((n+1)_3\) configuration from it. The main purpose of the article under review is the presentation of the subsequent characterization theorem:NEWLINENEWLINETHeorem. Let \((\Sigma,\Pi)\) be an \((n+1)_3\) configuration. Then \((\Sigma,\Pi)\) can be constructed by a \(1\)-point extension from an \(n_3\) configuration if and only if \((\Sigma,\Pi)\) is not one of the following configurations: {\parindent=0.7cm \begin{itemize}\item[(a)] the Fano configuration \item[(b)] the Pappus configuration \item[(c)] the Desargues configuration \item[(d)] a Fano-type configuration. NEWLINENEWLINE\end{itemize}} The author's proof is synthetical and rather long because an extensive case analysis has to be done. Finally, \(3\)-extensions are dealt with shortly.