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On the polar varieties of ruled hypersurfaces

Published online by Cambridge University Press:  09 November 2016

M.E. HERNANDES ,
R. MARTINS  and
M.E. RODRIGUES HERNANDES
M.E. HERNANDES
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790, Maringá-PR 87020-900, Brazil. e-mails: [email protected]; [email protected]; [email protected]
R. MARTINS
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790, Maringá-PR 87020-900, Brazil. e-mails: [email protected]; [email protected]; [email protected]
M.E. RODRIGUES HERNANDES
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790, Maringá-PR 87020-900, Brazil. e-mails: [email protected]; [email protected]; [email protected]
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Abstract

The aim of this work is to characterise the k-polar variety Pk(X) of an (n − 1)-ruled hypersurface X ⊂ ℂn+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 ℂn+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 Eu0(X) of X and, in addition, we exhibit (n − 1)-ruled hypersurfaces such that Eu0(X) = m, for any prescribed positive integer m.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 193 - 204
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

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