Let $X=\operatorname{Spec}k[x,y,z,w]/(xw-yz)\subset Y=\operatorname{Spec} k[x,y,z,w]$. Let $\tilde{Y}:=\operatorname{Blow}\_P Y$, where $P$ is the origin. The exceptional divisor is $E$.
Is the total transform of $X$ a variety with only double normal crossing singularity?



singularity of total transform
It is easy to see in coordinates. $\tilde{Y}$ in this scenario can be written as the vanishing of all $2\times 2$ minors of


$
\begin{bmatrix}
x & y & z & q \\
a\_0 & a\_1 & a\_2 & a\_3\end{bmatrix}
$


in $\mathbb{A}^4\times\mathbb{P}^3$. Then for example in the coordinate neighborhood $a\_0=1$ we have that the support of the total transform of $X$ can be written as $x\cdot ( a\_3-a\_2 a\_3)$, giving us the normal crossing (if we are not just considering the support then clearly the exceptional is non-reduced which prevents normal crossing).


