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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Rational surfaces and the canonical dimension of $ \operatorname{\mathbf{PGL}}_6$


Authors: J.-L. Colliot-Thélène, N. A. Karpenko and A. S. Merkur'ev
Translated by: the authors
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
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
Published electronically: June 27, 2008
MathSciNet review: 2381945
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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.


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Additional Information

J.-L. Colliot-Thélène
Affiliation: CNRS Mathématiques, UMR 8628, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, France
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
Email: merkurev@math.ucla.edu

DOI: https://doi.org/10.1090/S1061-0022-08-01021-2
Keywords: Algebraic group, projective linear group, rational surfaces, birational classification, canonical dimension
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
Dedicated: Dedicated to the 100th anniversary of the birth of Dmitriĭ Konstantinovich Faddeev
Article copyright: © Copyright 2008 American Mathematical Society