Blocking sets of Hermitian generalized quadrangles
A finite classical polar space \(\mathcal{P}\) of rank \(r \geq 2\) arises from the set of all absolute points and totally isotropic subspaces of a polarity of a projective space \(\mathrm{PG}(n, q)\), and \(r\) denotes the vector dimension of a maximal totally isotropic subspace of \(\mathcal{P}\). The Hermitian surface \(\mathcal{H}(3,q^2)\subset \mathrm{PG}(3,q^2)\) and the Hermitian variety \(\mathcal{H}(4, q^2 ) \subset \mathrm{PG}(4, q^2)\) are examples of classical generalized quadrangles, that is polar spaces of rank \(2\).NEWLINENEWLINELet \(B\) the set of totally isotropic \(k\)-dimensional subspaces of \(\mathcal{P}\). A blocking set of \(\mathcal{P}\) with respect to \(B\) is a set of points of \(\mathcal{P}\) that meets every element of \(B\); a blocking set is minimal if it does not contain a smaller blocking set.NEWLINENEWLINEThe authors construct infinite families of minimal blocking sets of the generalized quadrangles \(\mathcal{H}(3,q^2)\) and \(\mathcal{H}(4,q^2)\) with respect to lines. These examples do not lie in a hyperplane of \(\mathrm{PG}(3, q^2)\) and of \(\mathrm{PG}(4, q^2)\), respectively. All the examples of minimal blocking sets of \(\mathcal{H}(4,q^2)\) already known in the literature do not have this property.
scientific article; zbMATH DE number 6366560