Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings

TL;DR

This paper studies the structure of Rees algebras and reduced fiber cones of divisorial filtrations on two-dimensional normal local rings, providing explicit descriptions and new proofs of properties related to Noetherian conditions.

Contribution

It offers an explicit scheme-theoretic description of the Rees algebra and reduced fiber cone, and presents a new proof of their Noetherian properties in relation to the scheme structure.

Findings

The scheme structure of the Rees algebra is explicitly described.

The reduced fiber cone's scheme structure is characterized in terms of the analytic spread.

The paper proves that the Rees algebra is Noetherian iff the projective scheme is proper over R.

Abstract

Let $\mathcal I=\{I_n\}$ be a divisorial filtration on a two dimensional normal excellent local ring $(R,m_R)$. Let $R[\mathcal I]=\oplus_{n\ge 0}I_n$ be the Rees algebra of $\mathcal I$ and $\tau:\mbox{Proj}R[\mathcal I])\rightarrow \mbox{Spec}(R)$ be the natural morphism. The reduced fiber cone of $\mathcal I$ is the $R$-algebra $R[\mathcal I]/\sqrt{m_RR[\mathcal I]}$, and the reduced exceptional fiber of $\tau$ is $\mbox{Proj}(R[\mathcal I]/\sqrt{m_RR[\mathcal I]})$. We give an explicit description of the scheme structure of $\mbox{Proj}(R[\mathcal I])$. As a corollary, we obtain a new proof of a theorem of F. Russo, showing that $\mbox{Proj}(R[\mathcal I])$ is always Noetherian and that $R[\mathcal I]$ is Noetherian if and only if $\mbox{Proj}(R[\mathcal I])$ is a proper $R$-scheme. We give an explicit description of the scheme structure of the reduced exceptional fiber $\mbox{Proj}(R[\mathcal I]/\sqrt{m_RR[\mathcal I]})$ of $\tau$, in terms of the possible values 0, 1 or 2 of the analytic spread $\ell(\mathcal I)=\dim R[\mathcal I]/m_RR[\mathcal I]$. In the case that $\ell(\mathcal I)=0$, $\tau^{-1}(m_R)$ is the emptyset; this case can only occur if $R[\mathcal I]$ is not Noetherian.