L\"uroth's theorem for fields of rational functions in infinitely many permuted variables

Abstract

L\"uroth's theorem describes the dominant maps from rational curves over a field. In this note we study those dominant rational maps from cartesian powers $X^{\Psi}$ 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 $h:X^{\Psi}\xrightarrow{f^{\Psi}}Y^{\Psi}\to H\backslash Y^{\Psi}$, 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 describing the equivariant integral schemes over $X^{\Psi}$ of finite type is touched very briefly.