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

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 $\operatorname {ch} k = 3$ .