Spreads admitting regular elliptic covers (Q582296)

Let \(\Sigma =PG(3,q)\) denote projective 3-space over the finite field GF(q). A spread of \(\Sigma\) is any collection of \(q^ 2+1\) skew lines, necessarily partitioning the points of \(\Sigma\). By the well-known correspondence of \textit{J. André} [Math. Z. 60, 156-186 (1954; Zbl 0056.285)], every such spread determines a two-dimensional translation plane, and conversely every two-dimensional translation plane arises from such a spread. If S is any spread of \(\Sigma\), we say that S admits a regular elliptic cover provided that S contains q-1 pairwise disjoint reguli (partioning all but two fixed lines of S). Clearly, the regular spread as well as any André spread admits a regular elliptic cover. In this paper for odd \(q\geq 5\) the author constructs an infinite family of non-André spreads admitting regular elliptic covers. These are the only known non- André spread to admit such a cover. The collineation groups of these spreads are also discussed in detail.