A class of noncommutative projective surfaces

Let A = circle plus(i >= 0) A(i) 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)[l, l(-1); sigma], where sigma is an automorphism of the integral projective surface X. Then we prove that A can be written as a naive 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.