Singularities of the projective dual variety

Roland Abuaf

Vol. 253 (2011), No. 1, 1–17

DOI: 10.2140/pjm.2011.253.1

Abstract

Let X ⊂ ℙ^N be an irreducible, nondegenerate projective variety and let X^∗⊂ ℙ^N∗ be its projective dual. Let L ⊂ ℙ^N be a linear space such that ⟨L,T_X,x⟩≠ℙ^N for all x ∈ X_smooth and such that the lines in X meeting L do not cover X. If x ∈ X is general, we prove that the multiplicity of X^∗ at a general point of ⟨L,T_X,x⟩^⊥ is strictly greater than the multiplicity of X^∗ at a general point of L^⊥. This is a strong refinement of Bertini’s theorem.

Keywords

projective geometry, singularities, dual variety

Mathematical Subject Classification 2000

Primary: 14B05, 14N15

Milestones

Received: 20 July 2010

Revised: 10 March 2011

Accepted: 31 March 2011

Published: 28 November 2011

Correction: 21 June 2014