Lüroth's theorem for fields of rational functions in infinitely many permuted variables
- Authors: Rovinskii M.Z.1
-
Affiliations:
- Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, Moscow, Russia
- Issue: Vol 216, No 9 (2025)
- Pages: 86-113
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/309465
- DOI: https://doi.org/10.4213/sm10234
- ID: 309465
Cite item
Abstract
Lüroth's theorem describes the dominant maps from rational curves over a field. 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 $h\colon X^{\Psi}\xrightarrow{f^{\Psi}}Y^{\Psi}\to H\setminus Y^{\Psi}$, 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, produces all dominant equivariant maps from $X^{\Psi}$ in the case $\dim X=1$. For arbitrary $X$ some partial results are obtained.Also, a similar problem of the description of equivariant integral schemes over $X^{\Psi}$ of finite type is touched very briefly.
About the authors
Marat Zefirovich Rovinskii
Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, Moscow, Russia
Author for correspondence.
Email: marat@mccme.ru
Doctor of physico-mathematical sciences
References
- M. Demazure, “Sous-groupes algebriques de rang maximum du groupe de Cremona”, Ann. Sci. Ec. Norm. Super. (4), 3:4 (1970), 507–588
- D. M. Evans, P. R. Hewitt, “Continuous cohomology of permutation groups on profinite modules”, Comm. Algebra, 34:4 (2006), 1251–1264
- A. Grothendieck, J. Dieudonne, “Elements de geometrie algebrique. III. Etude cohomologique des faisceaux coherents. I”, Publ. Math. Inst. Hautes Etudes Sci., 11 (1961), 5–167
- M. Hanamura, “On the birational automorphism groups of algebraic varieties”, Compos. Math., 63:1 (1987), 123–142
- S. MacLane, “The universality of formal power series fields”, Bull. Amer. Math. Soc., 45:12 (1939), 888–890
- S. Montgomery, “Hopf Galois theory: a survey”, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., 16, Geom. Topol. Publ., Coventry, 2009, 367–400
- R. Nagpal, A. Snowden, Symmetric subvarieties of infinite affine space
- M. Rovinsky, “Motives and admissible representations of automorphism groups of fields”, Math. Z., 249:1 (2005), 163–221
- M. Rovinsky, “Semilinear representations of symmetric groups and of automorphism groups of universal domains”, Selecta Math. (N.S.), 24:3 (2018), 2319–2349
- J. H. Silverman, The arithmetic of elliptic curves, Grad. Texts in Math., 106, 2nd ed., Springer, Dordrecht, 2009, xx+513 pp.
- J. Tate, “Algebraic formulas in arbitrary characteristic”: S. Lang, Elliptic functions, Appendix 1, Grad. Texts in Math., 112, 2nd ed., Springer-Verlag, New York, 1987, 299–306
Supplementary files
