{'fraction_non_alphanumeric': 0.056856187290969896, 'fraction_numerical': 0.013377926421404682, 'mean_word_length': 3.9916666666666667, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the polar varieties of ruled hypersurfaces

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.