Теорема Люрота для полей рациональных функций от бесконечно многих переставляемых переменных

Математический сборник. 2025. Т. 216. № 9. С. 86–113.

Rovinsky M.

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
f Ψ
permutations of the factors X. At least some of such maps arise as compositions h : X Ψ −−→ Y Ψ →
f
H\Y Ψ , where X -−−→ Y is a dominant k-map and H is a group of birational automorphisms of Y |k,
acting diagonally on YΨ .
In characteristic 0, we show that this construction, when properly modified, gives all dominant
equivariant maps from XΨ, if dim X = 1. For arbitrary X, the results are only partial.
Also, a somewhat similar problem of describing the equivariant integral schemes over XΨ of
finite type is touched very briefly.

Keywords: алгебраические многообразия

Publication based on the results of: Geometry of algebraic varieties, derived categories, motives, and metrics of special holonomy (2025)