A focus on focal surfaces

2000/02/14 by E. Arrondo, Arrondo, E., M. Bertolini +3
Mathematics · #14N99 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14N99

paper · pdf · doi:10.48550/arxiv.math/0002109

Plain TeX, 33 pages with no figures

arxiv created 2000/02/14 · arxiv updated 2009/11/30

Abstract

We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal surface (degree, class, class of its hyperplane section, sectional genus and degrees of the nodal and cuspidal curve). We study in particular the congruences of chords to a smooth curve and the congruences of bitangents or flexes to a smooth surface. We find that they possess unexpected components in their focal surface, and conjecture that they are the only ones with this property.

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