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

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 $\tau :ProjR[\mathcal{I}])\to Spec(R)$ be the natural morphism. The reduced fiber cone of $\mathcal{I}$ is the $R$-algebra $R[\mathcal{I}]\mathrm{/}\sqrt{m_{R}R[\mathcal{I}]}$, and the reduced exceptional fiber of $\tau$ is $Proj(R[\mathcal{I}]\mathrm{/}\sqrt{m_{R}R[\mathcal{I}]})$. We give an explicit description of the scheme structure of $Proj(R[\mathcal{I}])$. As a corollary, we obtain a new proof of a theorem of F. Russo, showing that $Proj(R[\mathcal{I}])$ is always Noetherian and that $R[\mathcal{I}]$ is Noetherian if and only if $Proj(R[\mathcal{I}])$ is a proper $R$-scheme. We give an explicit description of the scheme structure of the reduced exceptional fiber $Proj(R[\mathcal{I}]\mathrm{/}\sqrt{m_{R}R[\mathcal{I}]})$ of $\tau$, in terms of the possible values 0, 1 or 2 of the analytic spread $\mathrm{\ell }(\mathcal{I})=\dim R[\mathcal{I}]\mathrm{/}{m}_{R}R[\mathcal{I}]$. In the case that $\mathrm{\ell }(\mathcal{I})=0$, ${\tau }^{-1}(m_R)$ is the emptyset; this case can only occur if $R[\mathcal{I}]$ is not Noetherian.