dual curve

《Algebraic Geometry》 · 第 54 页 · Chapter I，Exercise 7.3

Let $Y\subseteq\mathbb P^2$ be a curve. For each nonsingular point $P\in Y$, show that there is a unique line $T_P(Y)$ whose intersection multiplicity with $Y$ at $P$ is $>1$. This is the tangent line to $Y$ at $P$. Show that the mapping $P\mapsto T_P(Y)$ defines a morphism of $\operatorname{Reg}Y$ into $(\mathbb P^2)^*$. The closure of the image of this morphism is called the dual curve $Y^*\subseteq(\mathbb P^2)^*$ of $Y$.