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.

中文翻译：

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一个态射的 Bertini 不可约定理中的异常轨迹

我们基于有限域上的随机超平面切片，引入了一种在任意域上求解 Bertini 不可约性定理的新方法。扩展 Benoist 的结果，我们证明对于态射 $\phi \colon X \to \mathbb{P}^n$ 使得 $X$ 在几何上不可约且 $\phi$ 的非空纤维都具有相同的维数，超平面 $H$ 的轨迹使得 $\phi^{-1} H$ 不是几何上不可约的，其维度至多为 $\operatorname{codim} \phi(X)+1$。我们在超平面截面上给出了单向群的应用。