Asian Journal of Mathematics

Volume 9 (2005)

Number 4

Focal Loci in G(1,n)

Pages: 449 – 472

DOI: http://dx.doi.org/10.4310/AJM.2005.v9.n4.a1

Authors

Enrique Arrondo

Marina Bertolini

Cristina Turrini

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.

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