  One of the major open problems in noncommutative algebraic geometry is the
classification of noncommutative surfaces, and this paper resolves a
significant case of this problem. Specifically, let S denote the 3-dimensional
Sklyanin algebra over an algebraically closed field k and assume that S is not
a finite module over its centre. (This algebra corresponds to a generic
noncommutative P^2.) Let A be any connected graded k-algebra that is contained
in and has the same quotient ring as a Veronese ring S^(3n). Then we give a
reasonably complete description of the structure of A. This is most
satisfactory when A is a maximal order, in which case we prove, subject to a
minor technical condition, that A is a noncommutative blowup of S^(3n) at a
(possibly non-effective) divisor on the associated elliptic curve E. It follows
that A has surprisingly pleasant properties; for example it is automatically
noetherian, indeed strongly noetherian, and has a dualizing complex.
