The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism

Abstract

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 : X -> 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 codim phi(X) + 1. We give an application to monodromy groups above hyperplane sections.