Resolution of $D_n$ singularity and exceptional divisor

Question

Consider a $D_n$ singularity. Let's do type $D_4$ explicitly here, by defining as 0 set of $x^2+y^2z+z^3$ in $\mathbb{A}^3$. Then do a blowup, and consider the chart given by $u=x/z$, $v=y/z$ and $z$. Then in this chart, the blowup surface is given by $u^2+v^2z+z=0$. Then the exceptional divisor is defined by $u^2=0$, which should be a double curve. I think some of my understanding is wrong.