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Let I = {In} be a Q-divisorial filtration on a two dimensional normal excellent local ring (R, mR). Let R[I] = ⊕n≥0In be the Rees algebra of I and τ : ProjR[I]) → Spec(R) be the natural morphism. The reduced fiber cone of I is the R- algebra R[I]/ p mRR[I], and the reduced exceptional fiber of τ is Proj(R[I]/ p 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 τ −1 (mR) has only finitely many ir- reducible components. In Theorem 1.2, we give an explicit description of the scheme structure of Proj(R[I]). As a corollary, we obtain in Theorem 1.3 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. In Corollary 1.4 to Theorem 1.2, we give an explicit description of the scheme structure of the reduced exceptional fiber Proj(R[I]/ p mRR[I]) of τ , in terms of the possible values 0, 1 or 2 of the analytic spread l(I) = dim R[I]/mRR[I]. In the case that l(I) = 0, τ −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
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