The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism

We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $\phi \colon X \to \mathbb{P}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $\phi$ all have the same dimension, the locus of hyperplanes $H$ such that $\phi^{-1} H$ is not geometrically irreducible has dimension at most $\operatorname{codim} \phi(X)+1$. We give an application to monodromy groups above hyperplane sections.