Noetherian intersections of regular local rings of dimension two

Abstract

Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We describe the desingularization of projective models over D both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree Q(D). We prove that this describes all the Noetherian rings that are intersections of rings in Q(D).