Classifying birationally commutative projective surfaces

Let  be a Noetherian connected graded domain of Gelfand–Kirillov dimension 3 over an algebraically closed field. Suppose that the graded quotient ring of R is of the formwhere K is 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 the work of Rogalski and Stafford on birationally commutative projective surfaces generated in degree 1; our proof techniques are quite different.