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In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is: what are the minimal models within a birational class? It is not even clear a priori what the correct definition is of a minimal model in this context.
We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.
This is joint work in progress with Dan Rogalski and Toby Stafford.