# Is Normalization of log smooth scheme smooth?

Let $f:Y\rightarrow X$ be a finite flat morphism between smooth schemes over $Spec k$, where $k$ is a perfect field. Let $D$ be an irreducible and smooth divisor of $X$, $U=X\setminus D$ the complement of $D$. We assume that $V=Y\times_X U$ is also the complement of a smooth divisor $E$ of $Y$ and $V\rightarrow U$ is a finite etale morphism.

Let $\pi$ be a uniformizer of $D$ and $e\geq 1$ be an integer.

Let $X_1$ be a scheme smooth over $Z_e$ where $Z_e=X[T]/(T^e-\pi)$ if $e$ is invertible in $k$, or $Z_e=X[T,U^{\pm}]/(UT^e-\pi)$ if $e$ is not invertible in $k$. We assume that $D_1=(D\times_X X_1)_{red}$ is irreducible and $U_1$ is the complement of $D_1$.

Now we consider the normalization $Y_1$ of the scheme $Y\times_X X_1$ and put $V_1=V\times_U U_1$.

Question: Is $Y_1$ smooth over $Spec k$ ? If so, is $V_1$ also the complement of a smooth divisor of $Y_1$?

-