Ruled quintic surfaces in \(\mathrm{PG}(6, q)\) (Q1623009)

Let \(C\) be a non-degenerate conic in a plane contained in \(\mathrm{PG}(6,q)\). Let \(V\) be the set of points of \(\mathrm{PG}(6,q)\) lying on the \(q+1\) lines joining each point of \(C\) to the corresponding point under a specified projectivity to the twisted cubic directrix. The author shows that \(V\) is a variety of order 5 and dimension 2 and that all such scrolls are projectively equivalent. It is shown that \(C\) contains exactly \(q+1\) points and one non-degenerate conic. Using the Bruck-Bose setting, the author shows that \(V\) contains exactly \(q^2\) twisted cubics and that each can act as a directrix of \(V\). The number of 4 and 5 dimensional normal rational curves contained in \(V\) are counted as well as the number of 5-spaces that meet \(V\).