Focal loci in G(1,N)

Abstract

We introduce the different focal loci (focal points, planes and hyperplanes) of (n - 1)-dimensional families (congruences) of lines in P-n and study their invariants, geometry and the relation among them. We also study some particular congruences whose focal loci have special behaviour, namely (n - 1)-secant lines to an (n - 2)-fold and (n - 1)-tangent lines to a hypersurface. In case n = 4 we also give, under some smoothness assumptions, a classification result for these congruences.