The noncommutative geometry of elliptic difference equations

Understand

We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve.

• The construction agrees with one implicit in work of Van den Bergh (iterated blowups of noncommutative Hirzebruch surfaces), but the construction enables one to prove a number of new facts about these surfaces.
• We show that they are noncommutative smooth proper surfaces in the sense of Chan and Nyman, with projective Quot schemes, that moduli spaces of simple sheaves are Poisson and that moduli spaces classifying semistable sheaves of rank 0 or 1 are projective.
• We further show that the action of SL_2(Z) as derived autoequivalences of rational elliptic surfaces extends to an action as derived equivalences of surfaces in our family with K^2=0.