On invariant rational functions under rational transformations

• Jason Bell ^[1] ; Rahim Moosa ^[1] ; Matthew Satriano ^[1]
  1. [1] University of Waterloo

     University of Waterloo

     Canadá

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 3, 2024, págs. 1-23
• Idioma: inglés
• DOI: 10.1007/s00029-024-00940-8
• Enlaces
  • Texto completo
• Resumen

  • Let X be an algebraic variety equipped with a dominant rational map φ : X X. A new quantity measuring the interaction of (X, φ) with trivial dynamical systems is introduced; the stabilised algebraic dimension of (X, φ) captures the maximum number of new algebraically independent invariant rational functions on (X × Y , φ × ψ), as ψ : Y Y ranges over all dominant rational maps on algebraic varieties. It is shown that this birational invariant agrees with the maximum dim X where (X, φ) (X , φ )is a dominant rational equivariant map andφ is part of an algebraic group action on X . As a consequence, it is deduced that if some cartesian power of (X, φ) admits a nonconstant invariant rational function, then already the second cartesian power does.

• Referencias bibliográficas
  • Amerik, E., Bogomolov, F., Rovinsky, M.: Remarks on endomorphisms and rational points. Compos. Math. 147(6), 1819–1842 (2011)
  • Amerik, Ekaterina, Campana, Frédéric: Fibrations méromorphes sur certaines variétés à fibré canonique trivial. Pure Appl. Math. Q., 4(2, Special...
  • Atiyah, M. F., Macdonald, I. G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.,...
  • Bell, J., Rogalski, D., Sierra, S.J.: The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings. Israel J. Math. 180, 461–507...
  • Bell, Jason, Ghioca, Dragos, Reichstein, Zinovy: On a dynamical version of a theorem of Rosenlicht. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5),...
  • Blanc, Jérémy, Furter, Jean-Philippe: Topologies and structures of the Cremona groups. Ann. of Math. (2), 178(3):1173–1198, (2013)
  • Cantat, S.: Invariant hypersurfaces in holomorphic dynamics. Math. Res. Lett. 17(5), 833–841 (2010)
  • Cantat, S.: Morphisms between Cremona groups, and characterization of rational varieties. Compos. Math. 150(7), 1107–1124 (2014)
  • Chatzidakis, Z., Hrushovski, E.: Difference fields and descent in algebraic dynamics. I. J. Inst. Math. Jussieu 7(4), 653–686 (2008)
  • Chatzidakis, Z., Hrushovski, E.: Difference fields and descent in algebraic dynamics. II. J. Inst. Math. Jussieu 7(4), 687–704 (2008)
  • Chatzidakis, Z., Hrushovski, E.: Model theory of difference fields. Trans. Am. Math. Soc. 351, 2997–3071 (1999)
  • Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. 4(3), 507–588 (1970)
  • Grothendieck, Alexander: Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert. In Séminaire...
  • Jaoui, Rémi, Moosa, Rahim: Abelian reduction in differential-algebraic and bimeromorphic geometry. To appear in Annales de l’Institut Fourier....
  • Kamensky, Moshe: Definable groups of partial automorphisms. Selecta Math. (N.S.), 15(2):295–341, (2009)
  • Medvedev, Alice, Scanlon, Thomas: Invariant varieties for polynomial dynamical systems. Ann. of Math. (2), 179(1):81–177, (2014)
  • Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443 (1956)
  • The Stacks project authors. The stacks project. https://stacks.math.columbia.edu, 2023
  • Weil, A.: On algebraic groups of transformations. Am. J. Math. 77, 355–391 (1955)
  • Zhang, Shou-Wu: Distributions in algebraic dynamics. In Surveys in differential geometry. Vol. X, volume 10 of Surv. Differ. Geom., pages...