Projective dual variety ($Y^{\ast }\subset \mathbb{P}(V^{\ast })$)

For an irreducible projective variety $Y\subset \mathbb{P}(V)$, its projective dual is the closure of hyperplanes containing the embedded tangent space at some smooth point of $Y$. Over the complex numbers, biduality gives $(Y^{\ast }{)}^{\ast }=Y$. For a nondegenerate smooth curve in characteristic zero, the dual is an irreducible hypersurface; a general point corresponds to one ordinary tangency. Its defining incidence space consists of a point on the curve and a hyperplane containing its tangent line.

Over the complex numbers, an irreducible projective variety is recovered by applying projective dual variety twice. In its conormal incidence variety, pairs $(y,H)$ with $H$ tangent at $y$ become pairs $(H,y)$ under exchange of the two projective factors. At general smooth pairs, differentiating the incidence equation shows that the hyperplane defined by $y$ is tangent to the dual variety at $H$. Closure gives the same conormal variety with the factors exchanged, proving biduality. This is why the branch hypersurface of the Gauss map of a theta divisor determines a nonhyperelliptic canonical curve.