(Dual variety)
Let \(X \subseteq \PP^n\) be an irreducible projective variety, and let \(\dualof{(\PP^n)}\) be the projective space of hyperplanes in \(\PP^n\). A hyperplane \(H\) is tangent to \(X\) at a smooth point \(p\) when \(H\) contains the projective tangent space \(\mathbb{T}_p X\). The dual variety \(\dualof{X} \subseteq \dualof{(\PP^n)}\) is the closure of the set of hyperplanes tangent to \(X\) at some smooth point.
The incidence correspondence \[ \Phi = \overline{\ts{ (p, H) \st p \in X_{\mathrm{sm}},\ \mathbb{T}_p X \subseteq H }} \subseteq X \times \dualof{(\PP^n)} \] is irreducible of dimension \(n-1\), since over each smooth point \(p\) the fibre is a projective space of dimension \(n - 1 - \dim X\). So \(\dualof{X}\), the image of \(\Phi\) under the second projection, is irreducible of dimension at most \(n-1\). Over a field of characteristic zero, \(\dualof{(\dualof{X})} = X\).
If \(X \subseteq \PP^2\) is a smooth conic \(x^\top A x = 0\) with \(A\) symmetric and invertible, the tangent line at \(p\) has coordinates \(Ap\), and \(\dualof{X}\) is the conic \(\xi^\top A^{-1} \xi = 0\). A linear subspace \(\PP^k \subseteq \PP^n\) has dual variety the linear subspace of hyperplanes containing it, of dimension \(n-k-1\), so a dual variety need not be a hypersurface.
Dependencies
D-0SYCYSmooth and singular points; Jacobian and regularity criteriauses