The birational geometry of noncommutative surfaces

We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts from commutative geometry (blowups commute, Quot schemes are projective, etc.) carry over. The key new tool in studying these deformations is a relatively simple description of their derived categories and the relevant t-structures; this also allows us to establish nontrivial derived equivalences for deformations of elliptic surfaces. We also establish that the category of line bundles (suitably defined) on such a surface has a faithful representation in which the morphisms are difference or differential operators, and thus find that difference/differential equations can be viewed as sheaves on such surfaces. In particular, we find that many moduli spaces of sheaves on such surfaces have natural interpretations as moduli spaces of equations with (partially) specified singularities, and in particular find that the "isomonodromy" interpretation of discrete Painlev\'e equations and their generalizations has a natural geometric interpretation (twisting sheaves by line bundles).