# 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 $$\bigcup D_i$$? Does the number equal to the number of $$D_i$$'s? (i.e. $$D_i$$'s do not intersect with each other.) Thanks!

• It is bounded by the degree of the morphism. Feb 27, 2020 at 15:49
• But this is not sufficient... I expect this to be the number of $D_i$ or at least related to that... Feb 27, 2020 at 15:53
• No, the number of connected components does not equal the number of irreducible components. For example take $X \rightarrow \mathbf P^2$ a double cover branched over a smooth quartic curve $C$. For a line $L \subset \mathbf P^2$ that is bitangent to $C$ the inverse image of $L$ is a union of 2 intersecting copies of $\mathbf P^1$.
– Bort
Feb 27, 2020 at 16:18
• Yes, this is a good example. Feb 27, 2020 at 16:19
• If $f$ is a morphism of curves, there is no better bound than the degree of the morphism. Feb 27, 2020 at 17:36