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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Gorensteinness of the symbolic blow-ups for certain space monomial curves


Authors: Shiro Goto, Koji Nishida and Yasuhiro Shimoda
Journal: Trans. Amer. Math. Soc. 340 (1993), 323-335
MSC: Primary 13A30; Secondary 13H10, 14M05
MathSciNet review: 1124166
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Abstract: Let $ {\mathbf{p}} = {\mathbf{p}}({n_1},{n_2},{n_3})$ 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 = {T^{{n_1}}}$, $ Y = {T^{{n_2}}}$ , and $ Z = {T^{{n_3}}}$ with $ {\text{GCD}}({n_1},{n_2},{n_3}) = 1$. Then the symbolic Rees algebras $ {R_s}({\mathbf{p}}) = { \oplus _{n \geq 0}}{{\mathbf{p}}^{(n)}}$ are Gorenstein rings for the prime ideals $ {\mathbf{p}} = {\mathbf{p}}({n_1},{n_2},{n_3})$ with $ \min \{ {n_1},{n_2},{n_3}\} = 4$ and $ {\mathbf{p}} = {\mathbf{p}}(m,m + 1,m + 4)$ with $ m \ne 9,13$ . The rings $ {R_s}({\mathbf{p}})$ for $ {\mathbf{p}} = {\mathbf{p}}(9,10,13)$ and $ {\mathbf{p}} = {\mathbf{p}}(13,14,17)$ are Noetherian but non-Cohen-Macaulay, if $ \operatorname{ch}\,k = 3$ .


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1124166-4
PII: S 0002-9947(1993)1124166-4
Article copyright: © Copyright 1993 American Mathematical Society