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Izv. RAN. Ser. Mat., 2013, Volume 77, Issue 4, Pages 103–134 (Mi izv8021)  

This article is cited in 4 scientific papers (total in 4 papers)

Tori in the Cremona groups

V. L. Popovab

a Steklov Mathematical Institute of the Russian Academy of Sciences
b State University – Higher School of Economics

Abstract: We classify up to conjugacy all subgroups of certain types in the full, affine and special affine Cremona groups and prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results on the linearization problem by generalizing Białynicki-Birula's results of 1966–67 to disconnected groups. We prove fusion theorems for $n$-dimensional tori in the affine and special affine Cremona groups of rank $n$, and introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.

Keywords: Cremona group, affine Cremona group, algebraic torus, diagonalizable algebraic group, conjugate subgroups, fusion theorems, torsion primes.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00185-a
Ministry of Education and Science of the Russian Federation НШ-5139.2012.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations


DOI: https://doi.org/10.4213/im8021

Full text: PDF file (754 kB)
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English version:
Izvestiya: Mathematics, 2013, 77:4, 742–771

Bibliographic databases:

UDC: 512.745.4
MSC: 14E07, 14L17, 14R10
Received: 01.07.2012

Citation: V. L. Popov, “Tori in the Cremona groups”, Izv. RAN. Ser. Mat., 77:4 (2013), 103–134; Izv. Math., 77:4 (2013), 742–771

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. J. Déserti, “Some properties of the group of birational maps generated by the automorphisms of $\mathbb{P}_{\mathbb{C}}^{n}$ and the standard involution”, Math. Z., 281 (2015), 893–905  crossref  mathscinet  zmath  isi  elib  scopus
    2. Blanc J., Calabri A., “On Degenerations of Plane Cremona Transformations”, Math. Z., 282:1-2 (2016), 223–245  crossref  mathscinet  zmath  isi  scopus
    3. V. L. Popov, “Borel Subgroups of Cremona Groups”, Math. Notes, 102:1 (2017), 60–67  mathnet  crossref  crossref  mathscinet  isi  elib
    4. V. L. Popov, “Bass' triangulability problem”, Algebraic varieties and automorphism groups, Adv. Stud. Pure Math., 75, eds. K. Masuda, T. Kishimoto, H. Kojima, M. Miyanishi, M. Zaidenberg, Math. Soc. Japan, Tokyo, 2017, 425–441  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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