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The exceptional locus in the Bertini irreducibility theorem for a morphism

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 Added by Bjorn Poonen
 Publication date 2020
  fields
and research's language is English




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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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