Lüroth's theorem for fields of rational functions in infinitely many permuted variables
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915625918529536 |
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| author | Rovinsky, M. |
| author_facet | Rovinsky, M. |
| contents | Lüroth's theorem describes the dominant maps from rational curves over a field.
In this note we study those dominant rational maps from cartesian powers $X^Ψ$ of geometrically irreducible varieties $X$ over a field $k$ for infinite sets $Ψ$ that are equivariant with respect to all permutations of the factors $X$. At least some of such maps arise as compositions $h:X^Ψ\xrightarrow{f^Ψ}Y^Ψ\to H\backslash Y^Ψ$, where $X\xrightarrow{f}Y$ is a dominant $k$-map and $H$ is a group of birational automorphisms of $Y|k$, acting diagonally on $Y^Ψ$.
In characteristic 0, we show that this construction, when properly modified, gives all dominant equivariant maps from $X^Ψ$, if $\dim X=1$. For arbitrary $X$, the results are only partial.
Also, a somewhat similar problem of describing the equivariant integral schemes over $X^Ψ$ of finite type is touched very briefly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04028 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lüroth's theorem for fields of rational functions in infinitely many permuted variables Rovinsky, M. Algebraic Geometry Representation Theory Lüroth's theorem describes the dominant maps from rational curves over a field. In this note we study those dominant rational maps from cartesian powers $X^Ψ$ of geometrically irreducible varieties $X$ over a field $k$ for infinite sets $Ψ$ that are equivariant with respect to all permutations of the factors $X$. At least some of such maps arise as compositions $h:X^Ψ\xrightarrow{f^Ψ}Y^Ψ\to H\backslash Y^Ψ$, where $X\xrightarrow{f}Y$ is a dominant $k$-map and $H$ is a group of birational automorphisms of $Y|k$, acting diagonally on $Y^Ψ$. In characteristic 0, we show that this construction, when properly modified, gives all dominant equivariant maps from $X^Ψ$, if $\dim X=1$. For arbitrary $X$, the results are only partial. Also, a somewhat similar problem of describing the equivariant integral schemes over $X^Ψ$ of finite type is touched very briefly. |
| title | Lüroth's theorem for fields of rational functions in infinitely many permuted variables |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2408.04028 |