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

Abstract

Let p = p ( n 1 , n 2 , n 3 ) {\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 ] ] A = k[[X,Y,Z]] over a field k k defining the space monomial curve X = T n 1 X = {T^{{n_1}}} , Y = T n 2 Y = {T^{{n_2}}} , and Z = T n 3 Z = {T^{{n_3}}} with GCD ( n 1 , n 2 , n 3 ) = 1 {\text {GCD}}({n_1},{n_2},{n_3}) = 1 . Then the symbolic Rees algebras R s ( p ) = ⊕ n ≥ 0 p ( n ) {R_s}({\mathbf {p}}) = { \oplus _{n \geq 0}}{{\mathbf {p}}^{(n)}} are Gorenstein rings for the prime ideals p = p ( n 1 , n 2 , n 3 ) {\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3}) with min { n 1 , n 2 , n 3 } = 4 \min \{ {n_1},{n_2},{n_3}\} = 4 and p = p ( m , m + 1 , m + 4 ) {\mathbf {p}} = {\mathbf {p}}(m,m + 1,m + 4) with m ≠ 9 , 13 m \ne 9,13 . The rings R s ( p ) {R_s}({\mathbf {p}}) for p = p ( 9 , 10 , 13 ) {\mathbf {p}} = {\mathbf {p}}(9,10,13) and p = p ( 13 , 14 , 17 ) {\mathbf {p}} = {\mathbf {p}}(13,14,17) are Noetherian but non-Cohen-Macaulay, if ch k = 3 \operatorname {ch}\,k = 3 .