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

Let I={In} be a Q -divisorial filtration on a two dimensional normal excellent local ring (R,mR) . Let R[I]=circle plus n >= 0In be the Rees algebra of I and tau:ProjR[I])-> Spec(R) be the natural morphism. The reduced fiber cone of I is the R-algebra R[I]/mRR[I] , and the reduced exceptional fiber of tau is Proj(R[I]/mRR[I]) . In [7], we showed that in spite of the fact that R[I] is often not Noetherian, mRR[I] always has only finitely many minimal primes, so tau-1(mR) has only finitely many irreducible components. We give an explicit description of the scheme structure of Proj(R[I]) . As a corollary, we obtain a new proof of a theorem of F. Russo, showing that Proj(R[I]) is always Noetherian and that R[I] is Noetherian if and only if Proj(R[I]) is a proper R-scheme. We give an explicit description of the scheme structure of the reduced exceptional fiber Proj(R[I]/mRR[I]) of tau , in terms of the possible values 0, 1 or 2 of the analytic spread l(I)=dimR[I]/mRR[I] . In the case that l(I)=0 , tau-1(mR) is the emptyset; this case can only occur if R[I] is not Noetherian. At the end of the introduction, we give a simple example of a graded filtration J of a two dimensional regular local ring R such that Proj(R[J]) is not Noetherian. This filtration is necessarily not divisorial.