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 $τ:\mbox{Proj}R[\mathcal I])\rightarrow \mbox{Spec}(R)$ be the natural morphism. The reduced fiber cone of $\mathcal{I}$ is the $R$-algebra $R[\mathcal{I}]/\sqrt{m_{R}R[\mathcal{I}]}$, 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 $\ell (\mathcal{I})=\dim R[\mathcal{I}]/m_{R}R[\mathcal{I}]$. 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.

arXiv commentCorrected typos in introduction. 26 pages