I want to understand the process of determining simple/multiple points of projective curves.
Say $F(X,Y,Z)$ is a homogeneous polynomial; it defines a curve, and by abuse of language I will refer to this curve as $F$. Fulton's book says that the multiplicity of $F$ at $P$ of $Z$ is by definition the multiplicity of $F_\ast$ at $P$, where $F_\ast=F(X,Y,1$). But in the former case $P$ is a point in $\mathbb{P}^2$ whereas in the latter case $P$ is a point in $\mathbb{A}^2$. So I'm confused.
Could you write a simple example of determining simple/multiple points of a projective curve? You may assume I know how to determine such points on affine curves by Fulton's algorithm (see my previous question Multiplicity of a point on an affine curve that is not the origin)