On the polar varieties of ruled hypersurfaces (original) (raw)

Mathematical proceedings of the Cambridge Philosophical Society, 2016

Abstract

The aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurface X ⊂ ℂ n+1. More precisely, we prove that Pk(X) is empty for all k >1 and the first polar variety is empty or it is an (n − 2)-ruled variety in ℂ n+1, whose multiplicity is obtained by the multiplicity of the base curve and the multiplicity of one directrix of X. As a consequence we obtain the Euler obstruction Eu 0(X) of X and, in addition, we exhibit (n − 1)-ruled hypersurfaces such that Eu 0(X) = m, for any prescribed positive integer m.

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