This talk describes joint work with Tom Nevins. Noncommutative projective planes have been classified by Artin, Tate and Van den Bergh. In this talk we will show how one can construct (genuine commutative) projective moduli spaces for torsion-free sheaves on these noncommutative projective planes that are analogous to (indeed, deformations of) the moduli spaces of sheaves over the usual commutative projective plane P^2.
The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E inside P^2. In this case, the moduli space of line bundles over S with the appropriate invariants provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P^2-E. This is an ``elliptic'' analogue of earlier work of Berest, Wilson and others that showed that when S is the homogenised Weyl algebra the corresponding moduli space is Calogero-Moser space.