Rational surfaces and canonical dimension of PGL₆

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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 PGL₆ 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.