The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism
The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism
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态射贝尔蒂尼不可约定理中的特殊轨迹
DOI：
10.1093/imrn/rnaa182
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Kaloyan Slavov
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文献类型：
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作者：
B. Poonen;Kaloyan Slavov
我们引入了一种基于有限域上的随机超平面切片的任意域上贝尔蒂尼不可约定理的新方法。扩展Benoist的结果，我们证明对于态射$\phi \colon X \to \mathbb{P}^n$，使得$X$是几何不可约的，并且$\phi$的非空纤维都具有相同的维数，使得$\phi^{-1} H$不是几何不可约的超平面$H$的轨迹具有至多$\operatorname{codim} \phi(X)+1$的维数。我们给出了超平面截面之上的单峰群的应用。
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.