Spreads of nonsingular pairs in symplectic vector spaces
Consider the projective space \(\Sigma = \text{PG}(2n-1,F)\) of odd dimension \(2n-1 \geq 3\) over some field \(F\), equipped with a non-degenerate symplectic polarity \(\rho\). Such a polarity is induced by non-degenerate symplectic (alternating) form on the underlying \(2n\)-dimensional vector space over \(F\). A spread of \(\Sigma\) is a collection of mutually disjoint \((n-1)\)-spaces that partition the points of \(\Sigma\). An \((n-1)\)-space \(\Gamma\) of \(\Sigma\) is called totally isotropic if \(\Gamma^\rho=\Gamma\). A spread consisting of totally isotropic \((n-1)\)-spaces is often called a symplectic spread. In the paper under review the author defines an nsp-spread of \(\Sigma\) to be a spread consisting of non-singular pairs \(\{\Gamma,\Gamma^\rho\}\) (so \(\Gamma \cap \Gamma^\rho = \emptyset\)). Thus no spread element is an isotropic (or totally isotropic) space. It is shown that nsp-spreads exist whenever \(F\) is a finite field of odd characteristic or \(F\) is an algebraic number field. Moreover, restricting to \(n=2\), the number of nsp-spreads in PG\((3,q)\) is computed precisely for \(q=3,5,7\). The number of orbits of such spreads under the group PGSp\((4,q)\) is also computed for these values of \(q\). It should be noted that in [\textit{A. Cossidente, C. Culbert, G.~L. Ebert} and \textit{G. Marino}, ``On \(m\)-ovoids of \({\mathcal W}_3(q)\)'', Finite Fields Appl., in press] such spreads are called polarity-paired with respect to a symplectic polarity, where they are used to construct various \(m\)-ovoids for \(m=\frac{1}{2}(q+1)\) in the classical generalized quadrangle \(W_3(q)\).
scientific article; zbMATH DE number 5162055