Analytic spread of filtrations on two dimensional normal local rings

TL;DR

This paper extends classical results about the analytic spread of ideals to divisorial filtrations on two-dimensional normal local rings, revealing their asymptotic properties and Hilbert functions.

Contribution

It proves that McAdam's theorem applies to divisorial filtrations on two-dimensional normal excellent local rings and analyzes their Hilbert functions.

Findings

McAdam's theorem holds for divisorial filtrations in this setting.

Hilbert functions of fiber cones are sums of linear and bounded functions.

Asymptotic properties of divisors on resolutions are key to proofs.

Abstract

In this paper we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two dimensional normal excellent local ring $R$, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on $R$ has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of $R$. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.