 Projective dual variety

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.