On the geometry of Pr\"ufer intersections of valuation rings

TL;DR

This paper investigates the geometric structure of intersections of valuation rings, providing criteria to determine when such intersections form Pr"ufer domains using morphisms into the projective line.

Contribution

It introduces geometric criteria based on prime avoidance for identifying Pr"ufer domains among intersections of valuation rings, extending previous results.

Findings

Provides unified geometric criteria for Pr"ufer domain characterization.

Reduces the problem to prime avoidance questions.

Extends existing literature on valuation ring intersections.

Abstract

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is $A = \bigcap_{V \in Z}V$. All rings between $D$ and $F$ that are integrally closed in $F$ arise in such a way. Motivated by applications in areas such as multiplicative ideal theory and real algebraic geometry, a number of authors have formulated criteria for when $A$ is a Pr\"ufer domain. We give geometric criteria for when $A$ is a Pr\"ufer domain that reduce this issue to questions of prime avoidance. These criteria, which unify and extend a variety of different results in the literature, are framed in terms of morphisms of $Z$ into the projective line ${\mathbb{P}}^1_D$