Non-Cohen-Macaulay Symbolic Blow-ups for Space Monomial Curves

Abstract

Let p = p(nx, n2 , n3) 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"\ , Y = T"2, and Z = Tni with GCD(«,, n2, n3) = 1 .Then the symbolic Rees algebra Rs(p) = ©n>oP(n) f°r P = P(«2 + 2n + 2, n2 + 2n + 1, n2 + n + 1) is Noetherian but not Cohen-Macaulay if chk = p > 0 and n = pe with e > 1 .The same is true for p = p(n2, n2 + 1, n2 + n + 1) if ch k = p > 0 and n = pe > 3 .