Lüroth's theorem for fields of rational functions in infinitely many permuted variables

Abstract

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.