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We show that any commutative rationally ruled surface with a choice ofanticanonical curve admits a 1-parameter family of noncommutative deformationsparametrized by the Jacobian of the anticanonical curve, and show that manystandard facts from commutative geometry (blowups commute, Quot schemes areprojective, etc.) carry over. The key new tool in studying these deformationsis a relatively simple description of their derived categories and the relevantt-structures; this also allows us to establish nontrivial derived equivalencesfor deformations of elliptic surfaces. We also establish that the category ofline bundles (suitably defined) on such a surface has a faithful representationin which the morphisms are difference or differential operators, and thus findthat difference/differential equations can be viewed as sheaves on suchsurfaces. In particular, we find that many moduli spaces of sheaves on suchsurfaces 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 theirgeneralizations has a natural geometric interpretation (twisting sheaves byline bundles).

The birational geometry of noncommutative surfaces