Über Homomorphismen projektiver Räume. (On homomorphisms of projective spaces)
type: Item
instance of: scholarly article
title: Über Homomorphismen projektiver Räume. (On homomorphisms of projective spaces)
zbMATH Open document ID: 0655.51003
DOI: 10.1007/BF00147802
author: S. H. Smith
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51A10
zbMATH DE Number: 4069689
zbMATH Keywords: homomorphism of projective spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147802
OpenAlex ID: W2913734252
description: scientific article; zbMATH DE number 4069689