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

Abstract:

Let $\mathfrak {p} = \mathfrak {p}({n_1},{n_2},{n_3})$ denote the prime ideal in the formal power series ring $A = k[[X,Y,Z]]$ over a field $k$ defining the space monomial curve $X = {T^{{n_1}}},Y = {T^{{n_2}}}$, and $Z = {T^{{n_3}}}$ with $\operatorname {GCD} ({n_1},{n_2},{n_3}) = 1$. Then the symbolic Rees algebra ${R_s}(\mathfrak {p}) = { \oplus _{n \geq 0}}{\mathfrak {p}^{(n)}}$ for $\mathfrak {p} = \mathfrak {p}({n^2} + 2n + 2,{n^2} + 2n + 1,{n^2} + n + 1)$ is Noetherian but not Cohen-Macaulay if ${\text {ch}}k = p > 0$ and $n = {p^e}$ with $e \geq 1$. The same is true for $\mathfrak {p} = \mathfrak {p}({n^2},{n^2} + 1,{n^2} + n + 1)$ if ${\text {ch}}k = p > 0$ and $n = {p^e} \geq 3$ .