A note on quasi-Hermitian varieties and singular quasi-quadrics
Quasi-quadrics are introduced in [\textit{F. De Clerck, N. Hamilton, C. M. O'Keef} and \textit{T. Penttila}, Australas. J. Comb. 22, 151--166 (2000; Zbl 0970.51011)]. A set of points \(\mathcal H\) in \(PG(n,q^2)\) is called quasi-Hermitian variety, if it has the same intersection numbers with respect to hyperplanes as a non-singular Hermitian variety \(H(n-1,q^2)\) or as a cone with vertex a point and base a non-singular Hermitian variety \(H(n-2,q^2)\). The authors exhibit two constructions of quasi-Hermitian varieties which are not Hermitian varieties. A singular quasi-quadric is a point set \(\mathcal K\) in \(PG(n,q)\) which has the same numbers of points as a singular quadric \(Q\) and for which each intersection number with respect to hyperplanes is also an intersection number of \(Q\) with respect to hyperplanes. The authors prove the following theorem: Let \(\mathcal K\) be a set of \(q^3+q+1\) points in \(PG(4,q)\) such that every hyperplane intersects \(\mathcal K\) in \(q+1\), \(q^2+1\), or \(q^2+q+1\) points and designate a solid which intersects \(\mathcal K\) in \(q+1\) points as \textit{\((q+1)\)-solid}. If every \((q+1)\)-solid contains a line which meets \(\mathcal K\) in at least \(q\) points and if there are at least three \((q+1)\)-solids which intersect \(\mathcal K\) in a full line, then the set \(\mathcal K\) is a cone with vertex a point and base an ovoid.
scientific article; zbMATH DE number 5835851