Geometric configuration of integrally closed Noetherian domains

In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this end, we first describe the DVRs of $\mathbb{Q}(X)$ lying over ${\mathbb{Z}}_{(p)}$ for some prime $p\in \mathbb{Z}$, by distinguishing them according to whether the extension of the residue fields is algebraic or transcendental. We unify the known descriptions of such valuations by considering ultrametric balls in ${\mathbb{C}}_p$, the completion of the algebraic closure of the field ${\mathbb{Q}}_p$ of $p$-adic numbers. We then study when the intersection $R$ of such DVRs with $\mathbb{Q}[X]$ is of finite character, so that $R$ is a Krull domain, and we finally compute the divisor class group of $R$. It turns out that such a ring is formed by those polynomials which simultaneously map a finite union of ultrametric balls of ${\mathbb{C}}_p$ to its valuation domain ${\mathbb{O}}_p$, as $p\in \mathbb{Z}$ ranges through the set of primes. By a result of Heinzer, the Krull domains of this class are precisely the integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. This novel approach provides a geometric understanding of this class of integrally closed domains. Furthermore, we also describe the UFDs between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.