On Epimorphisms and Projectivities of Projective Planes1

Given an epimorphism φ: Π → Π' between projective planes Π and Π', it is an open question how the groups of projectivities of Π and Π' (regarded as permutation groups on projective lines) are related. Within this note we will not answer this sophisticated and hard problem in full, but we will address the question to which extend the projectivities of Π induce permutations on the lines of Π ' which are distinct from the projectivities of Π'. Questions of this kind are especially of interest when functions on projective planes which are invariant under perspectivities, such as orderings or half orderings, are subject to be lifted via an epimorphism. In particular, we show that for any pappian projective plane Π ' there exists a projective plane Π and an epimorphism φ: Π → †Π' such that the projectivities of Π induce the full symmetric group on the lines of Π' via φ (in this case no non-trivial function on Π' invariant under perspectivities lifts to Π). On the other hand, in terms of valutions and places of coordinatizing ternary fields we will characterize certain situations where only the projectivities of Π' and no further permutations are induced through φ.