Bertini irreducibility theorems over finite fields

Given a geometrically irreducible subscheme $ X \subseteq \mathbb{P}^n_{\mathbb{F}_q}$ of dimension at least $ 2$, we prove that the fraction of degree $ d$ hypersurfaces $ H$ such that $ H \cap X$ is geometrically irreducible tends to $ 1$ as $ d \to \infty $. We also prove variants in which $ X$ is over an extension of $ \mathbb{F}_q$, and in which the immersion $ X \to \mathbb{P}^n_{\mathbb{F}_q}$ is replaced by a more general morphism.National Science Foundation (U.S.) (Grant DMS-1069236