Jean-Louis Colliot-Thélène; Nikita A. Karpenko; Alexander S. Merkurjev: Rational surfaces and canonical dimension of PGL_6

Jean-Louis.Colliot-Thelene[at]math[.]u-psud[.]fr karpenko[at]math[.]jussieu[.]fr merkurev[at]math[.]ucla[.]edu

Submission: 2006, Nov 25

The "canonical dimension" of an algebraic group over a field by definition is the maximum of the canonical dimensions of principal homogenous spaces under that group. Over a field of characteristic zero, we prove that the canonical dimension of the projective linear group PGL_6 is 3. We give two distinct proofs, both of which rely on 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.

2000 Mathematics Subject Classification: 20G15, 14J26

Keywords and Phrases: Severi-Brauer varieties, canonical dimension, rational surfaces, del Pezzo surfaces

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