Independence of algebraic monodromy groups in compatible systems

In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

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.