Some noncommutative minimal surfaces

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra) as well as noncommutative analogues of P1×P1 and the more general Van den Bergh quadrics satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following result. Theorem A:Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and letA⊇Rbe any connected graded noetherian maximal order, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R. Let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules) of self-intersection(−1).