Example of morphism of irreducible varieties with reducible fibre

Question

Suppose $f: X \to Y$ is a morphism of irreducible affine or projective varieties. Does it then always follow that $f^{-1}(p)$ is irreducible for any point $p \in f(X)$? I think it's not true, and I'm trying to find a counterexample. Thank you!

Answer 1

A common example is $\operatorname{Spec}(\mathbb{C}[x,y,t] / \langle xy - t \rangle)$ over $\operatorname{Spec}(\mathbb{C}[t])$. Here, the fiber over $t=0$ is reducible.

Answer 2

(1) If $f:X\to Y$ is a finite map of varieties over a field of characteristic 0, with $\deg f=n>1$, then some fibers over a closed point will consist of $n$-many points. More concretely, $$Spec ~\mathbb{C}[x]/(x^n-1)\to Spec ~\mathbb{C}[x]$$ is a finite map with fibers over closed points consisting of $n$-points.

(2) Fancier example: take $g: Z=Bl_O(\mathbb{A}^2)\to \mathbb{A}^2$. Now blowup a closed point $p$ on the exceptional divisor $E$, to get $h: X=Bl_p(Z)\to Z$, then $f:X\to \mathbb{A}^2$ has fiber over the origin $O$, as union of two rational curves, so not irreducible.