2024/10/29 by Cutkosky, Steven Dale · 3 citations
#13A18 #14B04 #14B25 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.22512
Let \mathcal I=\In\ be a divisorial filtration on a two dimensional normal excellent local ring (R,mR). Let R[\mathcal I]=⊕n≥ 0In be the Rees algebra of \mathcal I and τ:ProjR[\mathcal I])→ Spec(R) be the natural morphism. The reduced fiber cone of \mathcal I is the R-algebra R[\mathcal I]/√(mRR[\mathcal I]), and the reduced exceptional fiber of τ is Proj(R[\mathcal I]/√(mRR[\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]/√(mRR[\mathcal I])) of τ, in terms of the possible values 0, 1 or 2 of the analytic spread ℓ(\mathcal I)=dim R[\mathcal I]/mRR[\mathcal I]. In the case that ℓ(\mathcal I)=0, τ-1(mR) is the emptyset; this case can only occur if R[\mathcal I] is not Noetherian.