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.
Abstract: We discuss projective families of lines of n, and in particular congruences of order one. After giving general results, we obtain a complete classification of the case of 4 in which there is a fundamental curve.
We discuss projective families of lines of ℙn, and in particular congruences of order one. After giving general results, we obtain a complete classification of the case of ℙ4 in which there is a fundamental curve.
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