Rational surfaces and the canonical dimension of 𝐏𝐆𝐋₆

Colliot-Thélène, J.-L.; Karpenko, N.; Merkur’ev, A.

doi:10.1090/S1061-0022-08-01021-2

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

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