On superheight conditions for the affineness of open subsets

In this paper we consider the open complement U of a hypersurface Y=V(a) in an affine scheme X. We study the relations between the affineness of U, the intersection of Y with closed subschemes, the property that every closed surface in U is affine, the property that every analytic closed surface is Stein and the superheight of the defining ideal a.