##### Seminar: Non-commutative minimal surfaces

###### Tue, May 16, 1:30pm – 2:30pm

Mike Artin has proposed that one should classify non-commutative projective surfaces (or connected graded domains of Gelfand-Kirillov dimension 3) in a manner similar to the commutative theory: one first classifies the minimal models within a given birational class and then show that any other surface can be blown down a finite number of times to reach such a minimal model.

The generic non-commutative P^2 is given by the Sklyanin algebra S and the minimal models birational to S are thought to be the Sklyanin algebra itself and Michel Van den Bergh’s quadrics. In this lecture we will show, using our non-commutative version of blowing down, that these algebras are minimal and indeed are minimal in a very strong sense: let R be a Sklyanin algebra or Van den Bergh quadric. Then any connected graded, noetherian over-ring of R with the same graded ring of fractions equals R.

This is joint work with Dan Rogalski and Sue Sierra.