A Class of Noncommutative Projective Surfaces

Let A = L i≥0 Ai be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A) = k(X)[t, t, σ], where σ is an automorphism of the integral projective surface X. Then we prove that A can be written as a näıve blowup algebra of a projective surface X birational to X. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in qgr-A will always be in (1-1) correspondence with the closed points of the scheme X.