Algebraic characterization of lineations on subsets of Desarguesian affine spaces (Q1919313)

A lineation is a map from a subset of a projective space into another projective space, which preserves collinearity of points. Let \(A,A'\) be desarguesian affine spaces, embedded in their respective projective closures. It is studied under which conditions a lineation \(\alpha\) from a subset \(S\) of \(A\) into \(A'\) can be extended to a lineation between the respective projective closures. It is shown that such an extension exists if \(A\) is a plane and \(S\) contains the union of a single point and three lines which intersect in three distinct points different from the first. Under mild hypotheses, which basicly exclude degenerate cases, and contain a certain ``local'' injectivity, it is shown that \(\alpha\) can be extended to an injective lineation of the respective projective closures, except for a very special case. This result is then generalized to arbitrary dimensions. The extension is possible only for a restriction of \(\alpha\) to a specified subset of \(S\). The hypotheses needed become a little technical here. In both cases, the extension is described algebraically by means of a monomorphism between the coordinatizing division rings. As a corollary, the well-known fundamental theorem of affine geometry is obtained. The main results can be viewed as a generalization of the fundamental theorem.