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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
DOI: https://doi.org/10.1090/S0002-9947-1993-1124166-4
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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Article copyright: © Copyright 1993 American Mathematical Society