I'm trying to find the simple/multiple points of the projective curve with the defining polynomial $F(X,Y,Z)=XZ-Y^2$. As far as I understand, I need to dehomogenize $F$ in each of the three standart charts $U_i$ and determine the simple/multiple of the resulting affine curve.
In the chart $U_0$ the dehomogenized polynomial is $G_0:=F(1,Y,Z)=Z-Y^2$, so all points on this affine curve have multiplicity one. This corresponds to the fact that all points $[1:Y:Z]\in V(XZ-Y^2)\subset \mathbb{P}^2$ are simple points of $F$.
The case of $U_2$ is similar: the dehomogenized polynomial is $G_2:=F(X,1,Z)=X-Y^2$, and so all points $[X:Y:1] \in V(XZ-Y^2)\subset \mathbb{P}^2$ are simple points of $F$.
In the case of $U_1$, the dehomogenized polynomial is $G_1:=F(X,1,Z)=XZ-1$. The point $(0,0)$ has multiplicity $0$, and the other points are simple. So all points $[X:1:Z] \in V(XZ-Y^2)\subset \mathbb{P}^2$ are simple points of $F$.
Is this correct? I'm unsure whether above is a full description of simple/multiple points.