Classifying orders in the Sklyanin algebra

Let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a nite module over its centre. (This algebra orresponds to a generic noncommutative P².) Let A = ⊕_i≥0Ai 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-eective) 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.