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

TL;DR

This work extends Lüroth-type descriptions to fields of rational functions in infinitely many permuted variables by analyzing invariant subfields of $F_{\\Psi}$ under the symmetric group $\rak{S}_{\Psi}$. In characteristic 0, for one-dimensional base varieties, every dominant $\rak{S}_{\Psi}$-equivariant map $X^{\Psi}\to Y^{\Psi}$ essentially factors through a base map $f:X\to Y$ and a diagonally acting birational group $H$, with higher dimensions remaining only partially resolved. The authors develop a toolbox based on $K$ähler$ $differentials and indecomposable injectives to classify invariant subfields, showing that for $ ext{tr.deg}(F|k)=1$ the invariant subfields are typically of the form $(L_{\Psi})^H$, and providing finite dimensional bounds on derivation algebras in the one-dimensional case. They also address symmetric irreducible subvarieties of $X^{\Psi}$, proving that invariant subvarieties must arise from diagonal embeddings or pullbacks via dominant maps, and discuss isotrivial finitely generated extensions, establishing conditions under which fixed fields are function fields over $k$. Overall, the paper links permutation symmetries to birational geometry and isotriviality phenomena, enriching Lüroth-type classifications for infinitely many permuted variables with structural insights and cohomological constraints.

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.

Theorems & Definitions (47)

• Lemma 2.1
• proof
• Conjecture 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Remark 2.5
• Lemma 3.1
• proof