Über Homomorphismen projektiver Räume. (On homomorphisms of projective spaces) (Q1109329)

The paper is based on the concept of homomorphism of projective spaces introduced by \textit{F. Machala} [Časopis Pěst. Mat. 100, 142-154 (1975; Zbl 0308.50011)]. These homomorphisms do not necessarily preserve dimensions. A homomorphism is called regular if for every three collinear image points there exist collinear preimage points. For regular homomorphisms, the following result of \textit{D. Row} [Bull. Aust. Math. Soc. 4, 155-158 (1971; Zbl 0204.535)] is shown to hold also for projective spaces: The epimorphism of a projective plane is already determined, up to isomorphism, by the preimage of one point. In analogy to projective planes, the concepts of a factor space and of the canonical epimorphism [\textit{S. Prieß-Crampe}, Geom. Dedicata 22, 21-37 (1987; Zbl 0609.51016)] are being introduced. The authors prove the necessity of the restrictions in their definitions by means of examples of homomorpisms in which collinear image points have no collinear preimages, and of homomorphisms mapping hyperplanes surjectively.