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# The Gorensteinness of the Symbolic Blow-Ups for Certain Space Monomial Curves

Shiro Goto, Koji Nishida and Yasuhiro Shimoda
Transactions of the American Mathematical Society
Vol. 340, No. 1 (Nov., 1993), pp. 323-335
DOI: 10.2307/2154559
Stable URL: http://www.jstor.org/stable/2154559
Page Count: 13
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## Abstract

Let p = p(n1, n2, n3) denote the prime ideal in the formal power series ring A = k[[ X, Y, Z]] over a field k defining the space monomial curve X = Tn1 , Y = Tn2 , and Z = Tn3 with GCD(n1, n2, n3) = 1. Then the symbolic Rees algebras $R_s(\mathbf{p}) = \bigoplus_{n \geq 0}\mathbf{p}^{(n)}$ are Gorenstein rings for the prime ideals p = p(n1, n2, n3) with min{n1, n2, n3} = 4 and p = p(m, m + 1, m + 4) with m ≠ 9, 13. The rings Rs(p) for p = p(9, 10, 13) and p = p(13, 14, 17) are Noetherian but non-Cohen-Macaulay, if $\operatorname{ch} k = 3$.

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