Lüroth's theorem for fields of rational functions in infinitely many permuted variables
M. Z. Rovinsky Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, Moscow, Russia
Abstract:
Lüroth's theorem describes the dominant maps from rational curves over a field.
We study those dominant rational maps from cartesian powers
$X^{\Psi}$ of geometrically irreducible varieties
$X$ over a field
$k$ for
infinite sets
$\Psi$ that are equivariant with respect to all permutations of the factors
$X$. At least some of such maps arise as compositions
$h\colon X^{\Psi}\xrightarrow{f^{\Psi}}Y^{\Psi}\to H\setminus Y^{\Psi}$, where
$X\xrightarrow{f}Y$ is a dominant
$k$-map and
$H$ is a group of birational automorphisms of
$Y|k$, acting diagonally on
$Y^{\Psi}$.
In characteristic
$0$ we show that this construction, when properly modified, gives all dominant equivariant maps from
$X^{\Psi}$ if
$\dim X=1$. For arbitrary
$X$ the results are only partial.
Also, a somewhat similar problem of the description of the equivariant integral schemes over
$X^{\Psi}$ of finite type is touched on very briefly.
Bibliography: 11 titles.
Keywords:
infinite symmetric group, infinite cartesian powers of algebraic varieties, equivariant maps of algebraic schemes.
MSC: 12E10,
14E05,
20C32 Received: 12.11.2024 and 29.11.2024
DOI:
10.4213/sm10234
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