Partial lineations between arguesian projective spaces
Over the last years the author and \textit{A. Fröhlicher} developed a new approach to the fundamental theorems of projective geometry, cf. their joint book [Modern projective geometry (Dordrecht) (2000; Zbl 0972.51001)]. In the present paper the author extends this approach to partial lineations between arguesian projective spaces. A partial lineation between projective spaces \(P\) and \(Q\) is a mapping \(g:P\setminus E \to Q\) defined outside a subset \(E \subset P\) which preserves collinearity and which maps every line that intersects \(E\) to a single point. The mapping \(g\) is called thick if there exists no line in \(P\) whose image consists of precisely two points, and \(g\) is called nondegenerate if it is thick and its image is not contained in a line. The author proves that every nondegenerate partial lineation between arguesian projective spaces is induced by a generalized semilinear map of the underlying vector spaces. Here, a generalized semilinear map is a map satisfying all the usual requirement of a semilinear map where a place is used instead of a field isomorphism. This theorem extends and sharpens several known results scattered in the literature.
fundamental theorems of projective geometry
partial lineations between arguesian projective spaces
scientific article; zbMATH DE number 1837316