On Epimorphisms and Projectivities of Projective Planes
by Franz Kalhoff
Abstract:
Given an epimorphism f from a projective plane P onto a
projective plane P', it is an open question how the groups of
projectivities of P and P' (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 P induce permutations on the lines of P' which are
distinct from the projectivities of P'. 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 P' there
exists a projective plane P and an epimorphism f from P onto P' such that
the projectivities of P induce the full symmetric group on the lines of P'
via f (in this case no non-trivial function on P' invariant under
perspectivities lifts to P). On the other hand, in terms of valutions and
places of coordinatizing ternary fields we will characterize certain
situations where only the projectivities of P' and no further permutations
are induced through f.
Key words and phrases: Epimorphisms of projective planes, the group of
projectivities, Fundamental Theorem of Projective Geometry, Pappian
projective planes, ternary fields, the radical of a ternary field, places
of ternary fields, valuations, uniform valuations.
1991 Mathematics Subject Classification: Primary 51A10; Secondary 51A30,
15A35, 17A60.
To appear in: Journal of Geometry
Contact: Franz.Kalhoff@Mathematik.Uni-Dortmund.DE