Let $X\to Y$ be a $d:1$ ramified covering map from a smooth complex projective variety $X$ to a smooth complex projective variety $Y$.

--What does the smoothness imply on the branch locus in $X$ or its image in $Y$? Does it imply that its a normal crossing?

--If not, assuming $Y=\mathbb{P}^n$, can we always deform $X$ to get a normal-crossing branch locus in $\mathbb{P}^n$ (or at least for its preimage in $X$)?

**Comment 1: Regarding the nice answer of Tony, you may assume $X$ is simply-connected.

**Comment 2: By normal crossing, I mean the reduced branch locus to be normal crossing.

Let $X\to Y$ be a $d:1$ ramified covering map from a smooth complex projective variety $X$ to a smooth complex projective variety $Y$.

Question 1. What does the smoothness imply on the branch locus in $X$ or its image in $Y$? Does it imply that its a normal crossing?

Question 2. If not, assuming $Y=\mathbb{P}^n$, can we always deform $X$ to get a normal-crossing branch locus in $\mathbb{P}^n$ (or at least for its preimage in $X$)?

Comment 1. Regarding the nice answer of Tony, you may assume $X$ is simply-connected.

Comment 2. By normal crossing, I mean the reduced branch locus to be normal crossing.

Let $X\to Y$ be a $d:1$ ramified covering map from a smooth complex projective variety $X$ to a smooth complex projective variety $Y$.

--What does the smoothness imply on the branch locus in $X$ or its image in $Y$? Does it imply that its a normal crossing?

--If not, assuming $Y=\mathbb{P}^n$, can we always deform $X$ to get a normal-crossing branch locus in $\mathbb{P}^n$ (or at least for its preimage in $X$)?

** Regarding the nice answer of Tony, you may assume $X$ is simply-connected**

Let $X\to Y$ be a $d:1$ ramified covering map from a smooth complex projective variety $X$ to a smooth complex projective variety $Y$.

--What does the smoothness imply on the branch locus in $X$ or its image in $Y$? Does it imply that its a normal crossing?

--If not, assuming $Y=\mathbb{P}^n$, can we always deform $X$ to get a normal-crossing branch locus in $\mathbb{P}^n$ (or at least for its preimage in $X$)?

**Comment 1: Regarding the nice answer of Tony, you may assume $X$ is simply-connected.

**Comment 2: By normal crossing, I mean the reduced branch locus to be normal crossing.