Hyperbolic fibrations of \(\text{PG}(3,q)\) (Q1279864)

A hyperbolic fibration of \(\text{PG}(3,q)\) is a set of \(q-1\) hyperbolic quadrics in \(\text{PG}(3,q)\) and two lines that together partition the point set of \(\text{PG}(3,q)\). For \(q\) odd, the authors review the known fibrations and construct a new family from a pencil of quadrics (containing a line as a reducible quadric) by replacing the set of \({1\over 2}(q+1)\) elliptic quadrics of the pencil by \({1\over 2}(q-1)\) hyperbolic quadrics and a line. Spreads can be obtained from a fibration by taking the two special lines, and by taking a regulus in each hyperbolic quadric. This way, the authors obtain new spreads from their new fibrations. They investigate the automorphism group of such spreads, and they conjecture that for \(q\geq 9\) these spreads contain only the \(q-1\) reguli of the hyperbolic quadrics of the fibration.