Let $f:X \to Y$ be a proper surjective morphism of quasi-projective noetherian schemes over $\mathbb{C}$. Assume that $Y$ is irreducible and $X$ is reduced, connected with finitely many irreducible components. Suppose further that every *closed* fiber of $f$ is an integral scheme. Is there any known additional condition on $f$ (other than flatness) under which we can conclude that $X$ is irreducible?



Proper morphism and irreducibility of schemes
This is false. Take for $X$ the union of the two axes in $\mathbf P^1\times\mathbf P^1$ and take for $f$ the first projection to $Y=\mathbf P^1$.


