On embeddable Minkowski planes (Q792630)

The classical example of a Minkowski plane is the points, rulings and non-degenerate conics on a ruled quadric \(Q_ 0\) in projective 3-space. For any point P not on \(Q_ 0\), an involution \(f_ p\) on \(Q_ 0\) is defined: if \(X\in Q_ 0\), then \(f_ p\) is \(X\leftrightarrow X'\), where X'\(\in P X\cap q_ 0\) and X'\(\neq X\) if \(| P X\cap Q_ 0| =2\). From this concept the authors abstract a structure (Q,R,R',F), where R and R' are two disjoint partitions of a set Q, with the property that every element of R meets every element of R' in a unique point. F is a family of involutions with properties in common with the involution \(f_ p\) on \(Q_ 0\). (Q,R,R',F) is called an abstract ruled quadric. The authors prove that every finite abstract ruled quadric Q with \(| Q| \geq 5\) is embeddable in PG(3,q), as a ruled quadric \(Q_ 0\) in which the involutions of F are involutions \(f_ p\) on \(Q_ 0\).