Equivalent definitions of Dual of a variety.

Question

Let $X$ be a closed irreducible projective variety of dimension $n$ in $\mathbb{P}_N.$ The dual of a projective variety is defined in following 2 ways. Def 1:If $X$ is nonsingular then the dual is the set of all hyperplanes of $\mathbb{P}_N$ which are tangent to $X.$ Def 2: let $A=\{(x,y)\in \mathbb{P}_N\times \mathbb{P}_N^*|x \ is\ nonsingular\ point\ of\ X \ and\ H_y\ is \ tangent \ to\ X\ at\ x\}.$ Here $H_y$ is hyperplane corresponding to $y.$ Take closure of $A$ and define dual to be image of second projection. How are these two definitions same for nonsingular varieties. I was reading the duality of projective spaces in the paper topology of complex projective varieties by Klaus Lamotke. The author says these two sets are the same, I dont see how? Any help is appreciated

Answer 1

The first definition is a bit sloppy. What does it mean for a hyperplane to be tangent to $X$ at a singular point? Do you include all possible hyperplanes through that point? Do you include only ones that are limits of tangent hyperplanes at nearby smooth points? The second definition does precisely the latter.

Consider, for example, the nodal curve $y^2z-x^2(x+z)=0$ in $\Bbb P^2$. What should happen at the origin?