Desmic systems of tetrahedra associated with a Morley-Petersen-study configuration (Q1266497)

Let \((a, a'), (b, b'), (c, c')\) be pairs of reciprocal polar lines with respect to a non-singular quadric \(\mathcal K\) in a 3-dimensional projective space \({\mathcal P}^3\) over an algebraically closed field. For a general choice of these line pairs, each of the four lines \((b, b', c, c'), (c, c', a, a'), (a, a', b, b')\) has a pair of common transversals denoted by \((d, d'), (e, e'), (f, f')\), respectively. If we let \((l, l'), (m, m'), (n, n')\) be the pairs of common transversals of the four lines \((a, a', d, d'), (b, b', e, e'), (c, c', f, f')\), respectively, then \((l, l'), (m, m')\) and \((n, n')\) have a common pair of transversals, denoted by \((s, s')\). Thus, to the initial general choice of the three line pairs, there are associated seven pairs of reciprocal polar lines with respect to \(\mathcal K\) (uniquely determined). A configuration of ten pairs of reciprocal polar lines of this kind is called a Morley-Petersen-Study configuration in \({\mathcal P}^3\) (an \({\mathcal M}-{\mathcal P}-{\mathcal S}\)). An interesting proof of the \({\mathcal M}-{\mathcal P}-{\mathcal S}\) configuration can be obtained by using the Klein correspondence between the lines of \({\mathcal P}^3\) and the points of Klein's quadric in \({\mathcal P}^5\). Using the Klein correspondence, the author proves that for a general choice of a pair of reciprocal polar lines with respect to the quadric \(\mathcal K\), five line congruences of class four and degree four are associated with an \({\mathcal M}-{\mathcal P}-{\mathcal S}\), such that each contains the chosen pair of polar lines and four pairs of reciprocal polar lines of the configuration. Furthermore, the intersection of the five congruences consists of eighteen lines which form the edges of two desmic systems of three tetrahedra. (Two tetrahedra are called desmic if each edge of one meets two opposite edges of the other).