Hostname: page-component-7f64f4797f-pmsz9 Total loading time: 0 Render date: 2025-11-05T18:42:34.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal subgroups in the Cremona group

Part of: Algebraic groups Linear algebraic groups and related topics Birational geometry

Published online by Cambridge University Press:  03 November 2025

Andrea Fanelli ,
Enrica Floris  and
Susanna Zimmermann Open the ORCID record for Susanna Zimmermann [Opens in a new window]
Andrea Fanelli
Affiliation:
Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France andrea.fanelli@math.u-bordeaux.fr
Enrica Floris
Affiliation:
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR 7348 du CNRS, Batiment H3 - Site du Futuroscope, 11 boulevard Marie et Pierre CURIE, TSA 61125, 86073 Poitiers Cedex 9, France, Institut Universitaire de France enrica.floris@univ-poitiers.fr
Susanna Zimmermann
Affiliation:
Mathematisches Institut, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland susanna.zimmermann@unibas.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for any $n\geq5$ there exist connected algebraic subgroups in the Cremona group $\text{Bir}(\mathbb P^n)$ that are not contained in any maximal connected algebraic subgroup. Our approach exploits the existence of stably rational, non-rational threefolds.

Keywords

Cremona groupsalgebraic groupsstable rationality

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations
Secondary: 14L30: Group actions on varieties or schemes (quotients) 20G99: None of the above, but in this section

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 9 , September 2025 , pp. 2442 - 2460
Check for updates
Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Beauville, A., Colliot-Thélène, J.-L., Sansuc, J.-J. and Swinnerton-Dyer, P., Variétés stablement rationnelles non rationnelles , Ann. of Math. (2) 121 (1985), 283–318; MR 786350.10.2307/1971174CrossRefGoogle Scholar
Blanc, J., Sous-groupes algébriques du groupe de Cremona, Transform. Groups 14 (2009), 249–285; MR 2504924.10.1007/s00031-008-9046-5CrossRefGoogle Scholar
Blanc, J., Cheltsov, I., Duncan, A. and Prokhorov, Y., Finite quasisimple groups acting on rationally connected threefolds, Math. Proc. Cambridge Philos. Soc. 174 (2023), 531–568.10.1017/S030500412200041XCrossRefGoogle Scholar
Blanc, J., Fanelli, A. and Terpereau, R., Automorphisms of $\mathbb{P}^1$ -bundles over rational surfaces , Épijournal Géom. Algébrique 6 (2022), 23; MR 4545346.Google Scholar
Blanc, J., Fanelli, A. and Terpereau, R., Connected algebraic groups acting on three-dimensional Mori fibrations , Int. Math. Res. Not. IMRN 2023 (2023), 1572–1689; MR 4537333.10.1093/imrn/rnab293CrossRefGoogle Scholar
Blanc, J. and Floris, E., Connected algebraic groups acting on Fano fibrations over $\mathbb{P}^1$ , Münster J. Math. 15 (2022), 1–46; MR 4476489.Google Scholar
Blanc, J. and Furter, J.-P., Topologies and structures of the Cremona groups , Ann. of Math. (2) 178 (2013), 1173–1198; MR 3092478.10.4007/annals.2013.178.3.8CrossRefGoogle Scholar
Blanc, J., Lamy, S. and Zimmermann, S., Quotients of higher-dimensional Cremona groups , Acta Math. 226 (2021), 211–318; MR 4281381.10.4310/ACTA.2021.v226.n2.a1CrossRefGoogle Scholar
Blanc, J., Schneider, J. and Yasinsky, E., Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank, Preprint (2022), arXiv:2211.17123v2.Google Scholar
Blanchard, A., Sur les variétés analytiques complexes , Ann. Sci. Éc. Norm. Supér. (3) 73 (1956), 157–202; MR 0087184.10.24033/asens.1045CrossRefGoogle Scholar
Brion, M., Invariants et covariants des groupes algébriques réductifs, Notes d’un cours à l’école d’été de Monastir (1996), http://www-fourier.univ-grenoble-alpes.fr/~mbrion/monastirrev.pdf.Google Scholar
Brion, M., Samuel, P. and Uma, V., Lecture on the structure of algebraic subgroups and geometric applications , CMI Lecture Series in Mathematics, vol. 1 (Hindustan Book Agency, New Delhi; Chennai Mathematical Institute (CMI), Chennai, 2013).Google Scholar
Cantat, S. and Lamy, S., Normal subgroups in the Cremona group , Acta Math. 210 (2013), 31–94, with an appendix by Yves de Cornulier; MR 3037611.10.1007/s11511-013-0090-1CrossRefGoogle Scholar
Demazure, M., Sous-groupes algébriques de rang maximum du groupe de Cremona (Algebraic subgroups of maximal rank in the Cremona group) , Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 507–588.10.24033/asens.1201CrossRefGoogle Scholar
Dolgachev, I. V. and Iskovskikh, V. A., Finite subgroups of the plane Cremona group , Progress in Mathematics, vol. 269 (Birkhäuser Boston, Boston, MA, 2009), 443–548; MR 2641179.Google Scholar
Enriques, F., Sui gruppi continui di trasformazioni cremoniane nel piano , Rendiconti Acc. Naz. Lincei (V), II (1893), 468–473.Google Scholar
Fogarty, J. and Norman, P., A fixed-point characterization of linearly reductive groups, in Contributions to Algebra, a collection of papers dedicated to Ellis Kolchin (Academic Press, 1977), 151–155.10.1016/B978-0-12-080550-1.50015-9CrossRefGoogle Scholar
Fong, P., Connected algebraic groups acting on algebraic surfaces , Ann. Inst. Fourier 14 (2023), 1–43.Google Scholar
Fong, P., Algebraic subgroups of the group of birational transformations of ruled surfaces , Épijournal Géom. Algébrique 7 (2023), 1–22.Google Scholar
Fong, P. and Zikas, S., Connected algebraic subgroups of groups of birational transformations not contained in a maximal one , C. R. Math. Acad. Sci. Paris 361 (2023), 313–322.10.5802/crmath.406CrossRefGoogle Scholar
Grothendieck, A., Géométrie formelle et géométrie algébrique. (Formal geometry and algebraic geometry) , Sem. Bourbaki 11 (1959), 28.Google Scholar
Hartshorne, R., Algebraic geometry , Graduate Texts in Mathematics, vol. 52 (Springer-Verlag, New York–Heidelberg, 1977); MR 0463157.Google Scholar
Hartshorne, R., Stable reflexive sheaves , Math. Ann. 254 (1980), 121–176.10.1007/BF01467074CrossRefGoogle Scholar
Kebekus, S., Boundedness results for singular Fano varieties, and applications to Cremona groups [following Birkar and Prokhorov-Shramov], Astérisque 422 (2020), exposé 1157, 253–290; MR 4224637.10.24033/ast.1136CrossRefGoogle Scholar
Kollár, J., Rational curves on algebraic varieties , in Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, vol. 32 (Springer-Verlag, Berlin, 1996).Google Scholar
Kollár, J., Automorphisms of unstable $\mathbb{P}^1$ -bundles, Preprint (2024), arXiv:2405. 18201.Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties (Cambridge University Press, Cambridge, 1998).10.1017/CBO9780511662560CrossRefGoogle Scholar
Lin, H.-Y. and Shinder, E., Motivic invariants of birational maps , Ann. of Math. (2) 199 (2023), 445–478.Google Scholar
Maruyama, M., On automorphism groups of ruled surfaces , J. Math. Kyoto Univ. 11 (1971), 89–112.Google Scholar
Matsumura, H., On algebraic groups of birational transformations , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 34 (1963), 151–155; MR 159825.Google Scholar
Moret-Bailly, L., Variétés stablement rationnelles non rationnelles , Astérisque 133-134 (1986), 223–236.Google Scholar
Mukai, S. and Umemura, H., Minimal rational threefolds, in Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol. 1016 (Springer, Berlin, 1983), 490–518; MR 726439.Google Scholar
Prokhorov, Y., p-elementary subgroups of the Cremona group of rank 3, in Classification of algebraic varieties, EMS Series Congress Reports (European Mathematical Society, Zürich, 2011), 327–338; MR 2779480.Google Scholar
Prokhorov, Y., Simple finite subgroups of the Cremona group of rank 3 , J. Algebraic Geom. 21 (2012), 563–600.10.1090/S1056-3911-2011-00586-9CrossRefGoogle Scholar
Prokhorov, Y. and Shramov, C., Jordan property for Cremona groups , Amer. J. Math. 138 (2016), 403–418; MR 3483470.10.1353/ajm.2016.0017CrossRefGoogle Scholar
Sarkisov, V. G., On conic bundle structures, Math. USSR Izvestiya 20 (1982), 355–390.10.1070/IM1983v020n02ABEH001354CrossRefGoogle Scholar
Serre, J.-P., Le groupe de Cremona et ses sous-groupes finis , Astérisque 332 (2010), exposé 1000, vii, 75–100; MR 2648675.Google Scholar
Shepherd-Barron, N. I., Stably rational irrational varieties , in The Fano conference (University of Torino, Turin, 2004), 693–700; MR 2112599.Google Scholar
Umemura, H., Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables , Nagoya Math. J. 79 (1980), 47–67.10.1017/S0027763000018924CrossRefGoogle Scholar
Umemura, H., Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type , Nagoya Math. J. 87 (1982), 59–78.10.1017/S0027763000019942CrossRefGoogle Scholar
Umemura, H., On the maximal connected algebraic subgroups of the Cremona group. I , Nagoya Math. J. 88 (1982), 213–246.10.1017/S0027763000020183CrossRefGoogle Scholar
Umemura, H., On the maximal connected algebraic subgroups of the Cremona group. II, in Algebraic groups and related topics (Kyoto/Nagoya, 1983), Advanced Studies in Pure Mathematics, vol. 6 (North-Holland, Amsterdam, 1985), 349–436; MR 803342.10.2969/aspm/00610349CrossRefGoogle Scholar
Umemura, H., Minimal rational threefolds. II. , Nagoya Math. J. 110 (1988), 15–80.10.1017/S0027763000002877CrossRefGoogle Scholar
Weil, A., On algebraic groups of transformations , Amer. J. Math. 77 (1955), 355–391; MR 0074083.10.2307/2372535CrossRefGoogle Scholar