Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3
over an uncountable algebraically closed field. Suppose that the graded
quotient ring of R is a skew-Laurent ring over a field; we say that R is a
birationally commutative projective surface. We classify birationally
commutative projective surfaces and show that they fall into four families,
parameterized by geometric data. This generalizes work of Rogalski and Stafford
on birationally commutative projective surfaces generated in degree 1; our
proof techniques are quite different.Comment: 60 pages; Proceedings of the LMS, 201
