The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism
<jats:title>Abstract</jats:title> <jats:p>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{...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Oxford University Press (OUP)
2021
|
| Online Access: | https://hdl.handle.net/1721.1/135257 |
| _version_ | 1811078058632806400 |
|---|---|
| author | Poonen, Bjorn Slavov, Kaloyan |
| author_facet | Poonen, Bjorn Slavov, Kaloyan |
| author_sort | Poonen, Bjorn |
| collection | MIT |
| description | <jats:title>Abstract</jats:title>
<jats:p>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.</jats:p> |
| first_indexed | 2024-09-23T10:52:43Z |
| format | Article |
| id | mit-1721.1/135257 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T10:52:43Z |
| publishDate | 2021 |
| publisher | Oxford University Press (OUP) |
| record_format | dspace |
| spelling | mit-1721.1/1352572021-10-28T04:25:12Z The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism Poonen, Bjorn Slavov, Kaloyan <jats:title>Abstract</jats:title> <jats:p>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.</jats:p> 2021-10-27T20:22:40Z 2021-10-27T20:22:40Z 2020 2021-05-25T18:52:39Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/135257 en 10.1093/IMRN/RNAA182 International Mathematics Research Notices Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press (OUP) MIT web domain |
| spellingShingle | Poonen, Bjorn Slavov, Kaloyan The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism |
| title | The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism |
| title_full | The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism |
| title_fullStr | The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism |
| title_full_unstemmed | The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism |
| title_short | The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism |
| title_sort | exceptional locus in the bertini irreducibility theorem for a morphism |
| url | https://hdl.handle.net/1721.1/135257 |
| work_keys_str_mv | AT poonenbjorn theexceptionallocusinthebertiniirreducibilitytheoremforamorphism AT slavovkaloyan theexceptionallocusinthebertiniirreducibilitytheoremforamorphism AT poonenbjorn exceptionallocusinthebertiniirreducibilitytheoremforamorphism AT slavovkaloyan exceptionallocusinthebertiniirreducibilitytheoremforamorphism |