Non-Cohen-Macaulay symbolic blow-ups for space monomial curves

Abstract

Let p = p ( n 1 , n 2 , n 3 ) \mathfrak {p} = \mathfrak {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 , Y = T n 2 X = {T^{{n_1}}},Y = {T^{{n_2}}} , and Z = T n 3 Z = {T^{{n_3}}} with GCD ⁡ ( n 1 , n 2 , n 3 ) = 1 \operatorname {GCD} ({n_1},{n_2},{n_3}) = 1 . Then the symbolic Rees algebra R s ( p ) = ⊕ n ≥ 0 p ( n ) {R_s}(\mathfrak {p}) = { \oplus _{n \geq 0}}{\mathfrak {p}^{(n)}} for p = p ( n 2 + 2 n + 2 , n 2 + 2 n + 1 , n 2 + n + 1 ) \mathfrak {p} = \mathfrak {p}({n^2} + 2n + 2,{n^2} + 2n + 1,{n^2} + n + 1) is Noetherian but not Cohen-Macaulay if ch k = p > 0 {\text {ch}}k = p > 0 and n = p e n = {p^e}