Hostname: page-component-7bb8b95d7b-nptnm Total loading time: 0 Render date: 2024-09-25T05:06:45.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal algebraic subgroups of the Cremona group of three variables—Imprimitive algebraic subgroups of exceptional type

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya, 464, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Enriques and Fano [4], [5] classified all the maximal connected algebraic subgroups of Cr3. Our aim is to give modern and rigorous proofs to their results. In [10], we studied the primitive subgroups. In this paper, we deal with exceptional imprimitive groups. The imprimitivity is an analytic notion. The natural translation of the imprimitivity in algebraic geometry is the de Jonquières type operation (definition (2.1)). Every de Jonquières type operation is imprimitive. However, the difference of these notions is subtle.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 87 , November 1982 , pp. 59 - 78
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[1] Blichfeldt, H. F., Finite collineation groups, The university of Chicago press, Chicago 1917.Google Scholar
[2] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 2 et 3, Hermann, Paris 1972.Google Scholar
[3] Clebsch, A., Theorie der binären algebraischen Formen, Teubner, Leipzig 1872.Google Scholar
[4] Enriques, F. e Fano, G., Sui gruppi di transformazioni cremoniane dello spazio, Annali di Matématica pura ed applicata, s. 2n 15 (1897), 59–98.Google Scholar
[5] Fano, G., I gruppi di Jonquières generalizzati, Atti della. R. Acc. di Torino, 33° (1898), 221–271.Google Scholar
[6] Lie, S., Theorie der Transformations gruppen, Teubner, Leipzig 1893.Google Scholar
[7] Matsumura, H., On algebraic groups of birational transformations, Lincei-R. Sc. fis. mat. e nat., 34 (1963), 151–155.Google Scholar
[8] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. of Math., 78 (1956), 401–443.Google Scholar
[9] Serre, J.-P., Algèbres de Lie semi-simples complexes, Benjamin, New York 1966.Google Scholar
[10] Umemura, H., Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables, Nagoya, Math., J., 79 (1980), 47–67.Google Scholar
[11] Weber, H., Lehrbuch der Algebra, Zweiter Band, Braunschweig 1899.Google Scholar