Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912102180978688 |
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| author | Cutkosky, Steven Dale |
| author_facet | Cutkosky, Steven Dale |
| contents | 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 $R[\mathcal I]/\sqrt{m_RR[\mathcal I]}$, and the reduced exceptional fiber of $τ$ 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 $τ$, in terms of the possible values 0, 1 or 2 of the analytic spread $\ell(\mathcal I)=\dim R[\mathcal I]/m_RR[\mathcal I]$. In the case that $\ell(\mathcal I)=0$, $τ^{-1}(m_R)$ is the emptyset; this case can only occur if $R[\mathcal I]$ is not Noetherian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings Cutkosky, Steven Dale Commutative Algebra Algebraic Geometry 14B04, 14B25, 13A18 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 $R[\mathcal I]/\sqrt{m_RR[\mathcal I]}$, and the reduced exceptional fiber of $τ$ 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 $τ$, in terms of the possible values 0, 1 or 2 of the analytic spread $\ell(\mathcal I)=\dim R[\mathcal I]/m_RR[\mathcal I]$. In the case that $\ell(\mathcal I)=0$, $τ^{-1}(m_R)$ is the emptyset; this case can only occur if $R[\mathcal I]$ is not Noetherian. |
| title | Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings |
| topic | Commutative Algebra Algebraic Geometry 14B04, 14B25, 13A18 |
| url | https://arxiv.org/abs/2410.22512 |