TITLE: Dual of a smooth hypersurface
QUESTION [5 upvotes]: Is it true that the dual of a smooth hypersurface $X$ of $\mathbb{P}^n$ of degree $d\ge 2$ is a hypersurface? If yes, could you give me a simple proof ? Or a reference?
Note that in this case, the dual is birational to $X$.

REPLY [5 votes]: A reference for this statement is Dolgachev's beautiful book Classical Algebraic Geometry, Chapter 1.