Remote Access St. Petersburg Mathematical Journal

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
Published electronically: June 27, 2008
MathSciNet review: 2381945
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 14L10, 14L15

Retrieve articles in all journals with MSC (2000): 14L10, 14L15

Additional Information

J.-L. Colliot-Thélène
Affiliation: CNRS Mathématiques, UMR 8628, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, France

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

A. S. Merkur'ev
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

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