Projective biduality theorem ($(Y^{\ast }{)}^{\ast }=Y$)

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.