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The branch locus in $Y$ need not be a normal crossings divisor even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether normalization we can find a finite morphsim $f : X \to \mathbb{P}^{2}$. The branch locus $B \subset \mathbb{P}^{2}$ will be a divisor by the Zariski-Nagata purity theorem but this divisor can never be normal crossings. If it was, then the fundamental group $\pi_{1}(\mathbb{P}^{2}-B)$ will be abelian by Fulton's nodal curve theorem. Since $\pi_{1}(X-f^{-1}(B)) \subset \pi_{1}(\mathbb{P}^{2}-B)$ is of finite index, and $\pi_{1}(X-f^{-1}(B))$ surjects onto $\pi_{1}(X)$ by say the the Fulton-Lazarsfeld connectedness theorem, it will follow that $\pi_{1}(X)$ is abelian which is a contradiction. In general the only implication of the smoothness will be the purity of branch loci: the branch locus has to be of pure dimension one.

The branch locus in $Y$ need not be a normal crossings divisor even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether normalization we can find a finite morphsim $f : X \to \mathbb{P}^{2}$. The branch locus $B \subset \mathbb{P}^{2}$ will be a divisor by the Zariski-Nagata purity theorem but this divisor can never be normal crossings. If it was, then the fundamental group $\pi_{1}(\mathbb{P}^{2}-B)$ will be abelian by Fulton's nodal curve theorem. Since $\pi_{1}(X-f^{-1}(B)) \subset \pi_{1}(\mathbb{P}^{2}-B)$ is of finite index, and $\pi_{1}(X-f^{-1}(B))$ surjects onto $\pi_{1}(X)$ e.g. by the Fulton-Lazarsfeld connectedness theorem, it will follow that $\pi_{1}(X)$ is abelian which is a contradiction. In general the only implication of the smoothness will be the purity of branch loci: the branch locus has to be of pure dimension one.

The branch locus in $Y$ need not be a normal crossings even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether normalization we can find a finite morphsim $f : X \to \mathbb{P}^{2}$. The branch locus $B \subset \mathbb{P}^{2}$ will be a divisor by the Zariski-Nagata purity theorem but this divisor can never be normal crossings. If it was, then the fundamental group $\pi_{1}(\mathbb{P}^{2}-B)$ will be abelian by Fulton's nodal curve theorem. Since $\pi_{1}(X-f^{-1}(B)) \subset \pi_{1}(\mathbb{P}^{2}-B)$ is of finite index, and $\pi_{1}(X-f^{-1}(B))$ surjects onto $\pi_{1}(X)$ by say the the Fulton-Lazarsfeld connectedness theorem, it will follow that $\pi_{1}(X)$ is abelian which is a contradiction. In general the only implication of the smoothness will be the purity of branch loci: the branch locus has to be of pure dimension one.

The branch locus in $Y$ need not be a normal crossings divisor even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether normalization we can find a finite morphsim $f : X \to \mathbb{P}^{2}$. The branch locus $B \subset \mathbb{P}^{2}$ will be a divisor by the Zariski-Nagata purity theorem but this divisor can never be normal crossings. If it was, then the fundamental group $\pi_{1}(\mathbb{P}^{2}-B)$ will be abelian by Fulton's nodal curve theorem. Since $\pi_{1}(X-f^{-1}(B)) \subset \pi_{1}(\mathbb{P}^{2}-B)$ is of finite index, and $\pi_{1}(X-f^{-1}(B))$ surjects onto $\pi_{1}(X)$ by say the the Fulton-Lazarsfeld connectedness theorem, it will follow that $\pi_{1}(X)$ is abelian which is a contradiction. In general the only implication of the smoothness will be the purity of branch loci: the branch locus has to be of pure dimension one.

The branch locus in $Y$ need not be a normal crossings even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether normalization we can find a finite morphsim $f : X \to \mathbb{P}^{2}$. The branch locus $B \subset \mathbb{P}^{2}$ will be a divisor by the Zariski-Nagata purity theorem but this divisor can never be normal crossings. If it was, then the fundamental group $\pi_{1}(\mathbb{P}^{2}-B)$ will be abelian by Fulton's nodal curve theorem. Since $\pi_{1}(X-f^{-1}(B)) \subset \pi_{1}(\mathbb{P}^{2}-B)$ is of finite index, and $\pi_{1}(X-f^{-1}(B))$ surjects onto $\pi_{1}(X)$ by say the the Fulton-Lazarsfeld connectedness theorem, it will follow that $\pi_{1}(X)$ is abelian which is a contradiction. IN general the only implication of the smoothness will be the purity of branch loci: the branch locus has to be of pure dimension one.

The branch locus in $Y$ need not be a normal crossings even when $Y$ is a projective space. Suppose $X$ is a smooth complex projective surface with non-abelian fundamental group. By Noether normalization we can find a finite morphsim $f : X \to \mathbb{P}^{2}$. The branch locus $B \subset \mathbb{P}^{2}$ will be a divisor by the Zariski-Nagata purity theorem but this divisor can never be normal crossings. If it was, then the fundamental group $\pi_{1}(\mathbb{P}^{2}-B)$ will be abelian by Fulton's nodal curve theorem. Since $\pi_{1}(X-f^{-1}(B)) \subset \pi_{1}(\mathbb{P}^{2}-B)$ is of finite index, and $\pi_{1}(X-f^{-1}(B))$ surjects onto $\pi_{1}(X)$ by say the the Fulton-Lazarsfeld connectedness theorem, it will follow that $\pi_{1}(X)$ is abelian which is a contradiction. In general the only implication of the smoothness will be the purity of branch loci: the branch locus has to be of pure dimension one.