We show first that a generic hypersurface V of degree d ≥ 3 in the complex projective space P^n of dimension n ≥ 3 has at least one hyperplane section V ∩H containing exactly n ordinary double points, alias A_1 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.
On the duals of smooth projective complex hypersurfaces / Dimca, Alexandru; Ilardi, Giovanna. - In: PUBLICACIONS MATEMÀTIQUES. - ISSN 0214-1493. - 68 (2024), 431{438 DOI: 10.5565/PUBLMAT6822404:(2024), pp. 431-438. [10.5565/PUBLMAT6822404]
On the duals of smooth projective complex hypersurfaces
Giovanna Ilardi
2024
Abstract
We show first that a generic hypersurface V of degree d ≥ 3 in the complex projective space P^n of dimension n ≥ 3 has at least one hyperplane section V ∩H containing exactly n ordinary double points, alias A_1 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.File | Dimensione | Formato | |
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