Ergänzung ''Über rationale Darstellungen von Kettengeometrien als projektive Varietäten''. (Complement to ''On rational representations of chain geometries as projective varieties'') (Q1081113)

In his paper in Geom. Dedicata 19, 287-293 (1985; Zbl 0573.51002) the author gave a very general construction for ''rational representations of chain geometries'' for a finite-dimensional K-algebra \({\mathfrak R}\), where the point set is a quasiprojective variety and the chains are smooth rational curves on it all of the same order r. In this construction a Grassmann map is followed by an appropriate projection. Here the author proves the following theorem: Let \({\mathfrak V}\) be the point set in a rational representation of the chain geometry \(\Sigma\) (K,\({\mathfrak R})\). Then \({\mathfrak V}\) is a projective variety if and only if \({\mathfrak R}\) is semisimple.