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arXiv · 2001.08672

The exceptional locus in the Bertini irreducibility theorem for a morphism

Bjorn Poonen·Kaloyan Slavov
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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 $\phi \colon X \to \mathbb{P}^n$ 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.

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Bjorn Poonen, Kaloyan Slavov. 2020-01-23. The exceptional locus in the Bertini irreducibility theorem for a morphism. https://doi.org/10.1093/imrn%2Frnaa182

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