On the Dimension and Degree of the Projective Dual Variety: A q-Analog of the Katz-Kleiman Formula

Abstract

Let X ⊂ P ^N be a complex projective variety. Let $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}\to {P} ^N$ be the projective space whose points are hyperplanes in P ^N. Let x ∈ X be any smooth point. A hyperplane $H \in \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}\to {P} ^N$ is said to be tangent to X at x if H contains the tangent subspace T _xX. The projective dual variety $X^V \subset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}\to {P} ^N$ is defined as the closure of the locus of those hyperplanes H which are tangent to X in some smooth point, see [7]. The name “projective dual” is justified by the biduality theorem [7]: the dual to X⋁ coincides with X.