  Let X be a projective surface, let \sigma be an automorphism of X, and let L
be a \sigma-ample invertible sheaf on X. We study the properties of a family of
subrings, parameterized by geometric data, of the twisted homogeneous
coordinate ring B(X, L, \sigma). In particular, we find necessary and
sufficient conditions for these subrings to be noetherian. We also study their
homological properties, their associated noncommutative projective schemes, and
when they are maximal orders. In the process, we produce new examples of
maximal orders; these are graded and have the property that no Veronese subring
is generated in degree 1.
  Our results are used in a companion paper to give defining data for a large
class of noncommutative projective surfaces.
