Complete spans on Hermitian varieties
Consider the projective geometry \(\text{PG}(3,q)\), where \(q\) is a prime power, and let \({\mathcal L}\) be a general line complex in \(\text{PG}(3,q)\). The usual extension of scalars from \(\text{GF}(q)\) to \(\text{GF}(q^2)\) leads to an extension of the lines of \({\mathcal L}\) to lines of \(\text{PG}(3,q^2)\). The authors show these extended lines cover a nonsingular Hermitian surface \(H\cong H(3,q^2)\) in \(\text{PG}(3,q^2)\). If this extension is applied to a symplectic spread \({\mathcal S}\) of \(\text{PG}(3,q)\), the extended lines of \({\mathcal S}\) form a complete \((q^2+ 1)\)-span of \(H\). Using the software package MAGMA the authors generate such spans for small values of \(a\). These examples indicate that the known bounds (upper and lower) for the size of a complete span of \(H(3,q^2)\) are weak. They conjecture that good bounds are probably quadratic in \(q\). In the last part of this article a generalization of the above process to the Hermitian variety \(H(5,q^2)\) is given. Since spreads do not exist in dimension 5, the authors start with a symplectic subgeometry \(W_5(q)\) of the point set of \(H(5,q^2)\). This subgeometry has a spread and extending scalars from \(\text{GF}(q)\) to \(\text{GF}(q^2)\) gives a complete \((q^3+1)\)-span of \(H(5,q^2)\).
scientific article; zbMATH DE number 1968852