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Flyingpanda
Flyingpanda
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Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do we know anything about the number of connected components of $\bigcup D_i$? Does the number equal to the number of $D_i$'s? (i.e. $D_i$'s do not intersect with each other.) Thanks!

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do we know anything about the number of connected components of $\bigcup D_i$? Thanks!

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do we know anything about the number of connected components of $\bigcup D_i$? Does the number equal to the number of $D_i$'s? (i.e. $D_i$'s do not intersect with each other.) Thanks!

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Flyingpanda
Flyingpanda
  • 393
  • 1
  • 7

Connected components of a codimension one fiber for a finite morphism

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do we know anything about the number of connected components of $\bigcup D_i$? Thanks!