On the discriminant locus of a generic projection

For a smooth projective variety $X\subseteq {\mathbb{P}}^N$ over an algebraically closed field of char $0$, we show that the discriminant locus of a generic projection of $X$ is projectively dual to a general linear section of the dual variety, and deduce a purity statement for the discriminant. Over $\mathbb{C}$, we also show that the fundamental group of the complement of the branch divisor arising from generic projection of a normal hypersurface surjects onto a braid group via braid monodromy.