On the duals of smooth projective complex hypersurfaces

• Dimca, Alexandru ^[1] ; Ilardi, Giovanna ^[2]
  1. [1] Université Côte d'Azur

     Université Côte d'Azur

     Arrondissement de Grasse, Francia

     
  2. [2] University of Naples Federico II

     University of Naples Federico II

     Nápoles, Italia

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 68, Nº. 2, 2024, págs. 431-438
• Idioma: inglés
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• Resumen

  • We first show that a generic hypersurface V of degree d ≥ 3 in the projective complex space P n of dimension n ≥ 3 has at least one hyperplane section V ∩H containing exactly n ordinary double points, alias A1 singularities, in general position, and no other singularities. Equivalently, the dual hypersurface V ∨ has at least one normal crossing singularity of multiplicity n. Using this result, we show that the dual of any smooth hypersurface with n, d ≥ 3 has at least a very singular point q, in particular a point q of multiplicity ≥ n.