Generalized ovals in PG\((3n-1,q)\), with \(q\) odd
A \textit{pseudo-oval} in PG\((3n-1,q)\), \(n\geq 1\), is a set of \(q^n+1\) distinct \((n-1)\)-dimensional subspaces such that any three of them generate PG\((3n-1,q)\). A particular property of a pseudo-oval \(O\) in PG\((3n-1,q)\) is that every \((n-1)\)-dimensional space \(\pi\) of this pseudo-oval is contained in a unique particular \((2n-1)\)-dimensional subspace, called the \textit{tangent space } of \(O\) at \(\pi\). The classical example of a pseudo-oval is the \textit{pseudo-conic } which is equal to the conic of PG\((2,q^n)\), but represented over \(\mathbb{F}_q\). The author presents in this article a characterization result on pseudo-ovals of PG\((3n-1,q)\), \(q\) odd: Let \(O=\{\pi_0,\dots,\pi_{q^n}\}\) be a pseudo-oval of PG\((3n-1,q)\), \(q\) odd, and let \(\tau_i\) be the tangent space at the space \(\pi_i\). If for any three distinct \(i,j,k\), with \(0\leq i,j,k\leq q^n\), the triples \(\{\pi_i,\pi_j,\pi_k\}\) and \(\{\tau_i,\tau_j,\tau_k\}\) are in perspective, then \(O\) is a pseudo-conic, and the converse also holds. This theorem has equivalent formulations in terms of generalized quadrangles and Laguerre planes. An important theorem in finite geometry is the \textit{Lemma of tangents} of B. Segre. The author presents a similar \textit{Lemma of tangents} for generalized ovals in PG\((5,q)\), \(q\) odd. He then uses this new lemma of tangents to prove the following theorem: Let \(O=\{L_0,\dots,L_{q^2}\}\) be a generalized oval in PG\((5,q)\), with \(q\) odd, and let \(\tau_i\) be the tangent space of \(O\) at \(L_i\). Then for any three distinct \(i,j,k\), with \(0\leq i,j,k\leq q^2\), there is either a non-singular quadric containing \(L_i,L_j,L_k\) and having \(\tau_i,\tau_j,\tau_k\) as tangent spaces at respectively \(L_i,L_j,L_k\), or \(\{L_i,L_j,L_k\}\) and \(\{\tau_i,\tau_j,\tau_k\}\) are in semi-perspective but not in perspective.
scientific article; zbMATH DE number 5942235