Rational surfaces and the canonical dimension of PGL_6

摘要

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 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.