Focal Loci in G(1,n)

Abstract

We introduce the different focal loci (focal points, planes and hyperplanes) of \break (n-1)-dimensional families (congruences) of lines in ${\Bbb P}^{n}$ and study their invariants, geometry and the relation among them. We also study some particular congruences whose focal loci have special behavior, 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.