Rational surfaces and the canonical dimension of $\operatorname {\mathbf {PGL}}_6$
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J.-L. Colliot-Thélène, N. A. Karpenko and A. S. Merkur’ev
Translated by: the authors - St. Petersburg Math. J. 19 (2008), 793-804
- DOI: https://doi.org/10.1090/S1061-0022-08-01021-2
- Published electronically: June 27, 2008
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Abstract:
By definition, the “canonical dimension” of an algebraic group over a field is the maximum of the canonical dimensions of the principal homogeneous spaces under that group. Over a field of characteristic zero, it is proved that the canonical dimension of the projective linear group $\operatorname {\mathbf {PGL}}_6$ is 3. We give two different proofs, both of which lean upon the birational classification of rational surfaces over a nonclosed field. One of the proofs involves taking a novel look at del Pezzo surfaces of degree 6.References
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Bibliographic Information
- J.-L. Colliot-Thélène
- Affiliation: CNRS Mathématiques, UMR 8628, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, France
- MR Author ID: 50705
- Email: Jean-Louis.Colliot-Thelene@math.u-psud.fr
- N. A. Karpenko
- Affiliation: Université Pierre et Marie Curie – Paris 6, Institut de Mathématiques de Jussieu, 4 place Jussieu, F-75252 Paris Cedex 05, France
- Email: karpenko@math.jussieu.fr
- A. S. Merkur’ev
- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
- MR Author ID: 191878
- ORCID: 0000-0002-4447-1838
- Email: merkurev@math.ucla.edu
- Received by editor(s): September 17, 2007
- Published electronically: June 27, 2008
- Additional Notes: This paper is the outcome of a discussion during a hike at Oberwolfach
- © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 793-804
- MSC (2000): Primary 14L10, 14L15
- DOI: https://doi.org/10.1090/S1061-0022-08-01021-2
- MathSciNet review: 2381945
Dedicated: Dedicated to the 100th anniversary of the birth of Dmitriĭ Konstantinovich Faddeev