# Irreducibility of fiber product of irreducible varieties via dominant morphisms

Let $X,Y,Z$ are irrreducible varieties. $f:X\to Y$ is prpoer surjective and $g:Z \to Y$ is dominant.

Then, $X\times_Y Z$ is irreducible?

Moreover, it will be very helpful for me if there are other conditions of morphisms $f,g$ that makes $X\times_Z Y$ irreducible.

• There are easy counterexamples where $X$, $Y$, and $Z$ are fields and the morphisms are finite Galois extensions. For example, $\mathbb C \otimes_{\mathbb R} \mathbb C \cong \mathbb C \times \mathbb C$. You can make more geometric examples by taking finite coverings between curves. – R. van Dobben de Bruyn Apr 15 '17 at 5:01
• There are much stranger counterexamples where the morphisms even have geometrically integral fibres. For example, consider a non-small blow-up like $\operatorname{Bl}_p(\mathbb P^3) \to \mathbb P^3$. Take the product of this morphism $X \to Y$ with itself, and note that its fibre above $y \in Y$ is $X_y \times X_y$. Above $p$, we get $\mathbb P^2 \times \mathbb P^2$, and every other fibre is an isomorphism. Thus, $X \times_Y X$ has components of different dimensions! – R. van Dobben de Bruyn Apr 15 '17 at 5:16

A good criterion for the irreducibility of a fiber product is when one of the morphisms, say $f$, is flat with irreducible fibers. Flatness implies that $X\times_YZ\to Z$ is also flat. Hence every irreducible component maps dominantly to $Z$. The second condition implies that there is only one component of that type.