Essential dimension relative to branched covers of degree at most n
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909870777696256 |
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| author | Farb, Benson Wolfson, Jesse |
| author_facet | Farb, Benson Wolfson, Jesse |
| contents | We prove for various finite groups $G$ and integers $n\geq 1$ that there are families of equations with Galois group $G$ that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most $n$. In more geometric language, there are $G$-varieties $X$ with the following property: for any $G$-equivariant branched cover $\widetilde{X}\to X$ of degree $\leq n$, there is no dominant rational $G$-map $\widetilde{X}\dashrightarrow C$ to any $G$-curve $C$. The method of proof is new, and applies in cases where previous methods do not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22786 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Essential dimension relative to branched covers of degree at most n Farb, Benson Wolfson, Jesse Algebraic Geometry Number Theory We prove for various finite groups $G$ and integers $n\geq 1$ that there are families of equations with Galois group $G$ that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most $n$. In more geometric language, there are $G$-varieties $X$ with the following property: for any $G$-equivariant branched cover $\widetilde{X}\to X$ of degree $\leq n$, there is no dominant rational $G$-map $\widetilde{X}\dashrightarrow C$ to any $G$-curve $C$. The method of proof is new, and applies in cases where previous methods do not. |
| title | Essential dimension relative to branched covers of degree at most n |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2510.22786 |