Blowing up finitely supported complete ideals in a regular local ring

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• 作者：William Heinzer^a ; ^{heinzer@purdue.edu" class="auth_mail" title="E-mail the corresponding author} ; Youngsu Kim^b ; ^{youngsu.kim@ucr.edu" class="auth_mail" title="E-mail the corresponding author} ; Matthew Toeniskoetter^a ; ^{mtoenisk@purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13A30 ; 13C05 ; secondary ; 13E05 ; 13H05
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：458
• 期：Complete
• 页码：364-386
• 全文大小：482 K

文摘

Let I   be a finitely supported complete m$\mathfrak{m}$-primary ideal of a regular local ring (R,m)$(R,\mathfrak{m})$. We consider singularities of the projective models $\mathrm{Proj}R[It]$ and $\mathrm{Proj}\overline{R[It]}$ over Spec R  , where $\overline{R[It]}$ denotes the integral closure of the Rees algebra R[It]$R[It]$. A theorem of Lipman implies that the ideal I   has a unique factorization as a ⁎-product of special ⁎-simple complete ideals with possibly negative exponents for some of the factors. If $\mathrm{Proj}\overline{R[It]}$ is regular, we prove that $\mathrm{Proj}\overline{R[It]}$ is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke–Sally in dimension 2, we prove that every local ring S   on $\mathrm{Proj}\overline{R[It]}$ that is a unique factorization domain is regular. Moreover, if dim⁡S≥2$\dim S\ge 2$ and S dominates R, then S is an infinitely near point to R, that is, S is obtained from R by a finite sequence of local quadratic transforms.