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

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.