A characterization of quadrics by intersection numbers
The paper contains a characterization for quadrics in projective spaces \(\mathrm{PG}(n, q)\), \(n\geq 4\), by intersection numbers with planes (2-dimensional projective subspaces) and solids (3-dimensional projective subspaces). First, a corollary of a theorem of \textit{O. Ferri} and \textit{G. Tallini} [Rend. Mat. Appl., VII. Ser. 11, No. 1, 15--21 (1991; Zbl 0724.51016)] is proved. Such a corollary reads as follows. Theorem. If a set of points \(K\) in \(\mathrm{PG}(4, q)\) is such that it intersects all planes in \(1, q+1\) or \(2 q + 1\) points and all solids in \(q^2+1, q^2+q+1\) or \(q^2+2q+1\) points, then \(K\) is a parabolic quadric of \(\mathrm{PG}(4, q)\). Then, this corollary is extended to all quadrics of \(\mathrm{PG}(n, q)\), \(n\geq 4\), and the following theorem is proved. Theorem. If a set of points \(K\) in \(\mathrm{PG}(n, q)\), \(n\geq 4\) intersects planes and solids in the same numbers of points as quadrics, then \(K\) is one of the following: {\parindent11mm \begin{itemize}\item[(i)] the projective space \(\mathrm{PG}(n, q)\), \item[(ii)] a hyperplane in \(\mathrm{PG}(n, q)\), \item[(iii)] a quadric in \(\mathrm{PG}(n, q)\). \item[(iv)] for \(q\) even, \item[(iv] a cone with vertex an \((n - 3)\)-dimensional space and base an oval. \item[(iv] a cone with vertex an \((n - 4)\)-dimensional space and base an ovoid. \end{itemize}}
scientific article; zbMATH DE number 5536727
scientific article