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

TL;DR

This paper extends L"uroth's theorem to describe dominant equivariant rational maps from infinite Cartesian powers of varieties, especially for curves, revealing their structure via compositions involving automorphism groups.

Contribution

It characterizes all dominant equivariant maps from infinite products of curves over fields of characteristic zero, generalizing classical L"uroth's theorem to a new infinite-dimensional setting.

Findings

For curves, all equivariant maps are compositions involving dominant maps and automorphism groups.

Partial results are obtained for higher-dimensional varieties.

The paper briefly discusses equivariant integral schemes over these infinite products.

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.