Noetherian intersections of regular local rings of dimension two

TL;DR

This paper investigates the structure of Noetherian rings formed by intersecting overrings within the quadratic tree of a 2-dimensional regular local ring, linking algebraic desingularization with order-theoretic properties.

Contribution

It provides a detailed description of the desingularization process and characterizes Noetherian intersections in terms of the quadratic tree structure.

Findings

Characterization of Noetherian intersections in quadratic trees

Connection between desingularization and saturation of ideals

Order-theoretic description of projective model desingularization

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 examine the Noetherian rings that are intersections of rings in $Q(D)$. To do so, 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 $Q(D)$.