This article presents the theory of focal locus applied to the hypersurfaces in the projective space which are (finitely) covered by linear spaces and such that the tangent space is constant along these spaces.
We prove that the general fibre of the i-th Gauss map has dimension m if and only if at the general point the (i+1)-th fundamental form consists of cones with vertex a fixed Pm−1, extending a known theorem for the usual Gauss map. We prove this via a recursive formula for expressing higher fundamental forms. We also show some consequences of these results.
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SYNAT - “Interdisciplinary System for Interactive Scientific and Scientific-Technical Information”.