Some necessary and sufficient condition for finite generation of symbolic Rees rings (Q2682813)

Let \(A\) be a commutative Noetherian ring and \(P \subset A\) a prime ideal. The \(n\)th symbolic power of \(P\) is defined to be \(P^{(n)} = P^nA_P \cap A\), that is, the \(P\)-primary component of \(P^n\). Many authors are interested in the Noetherian property of the symbolic Rees algebra \(\bigoplus_{n \geq 0} P^{(n)} t^n\), for example [\textit{S. Goto} et al., Proc. Am. Math. Soc. 120, No. 2, 383--392 (1994; Zbl 0796.13005)]. In the present paper, the authors studied this question when \(P\) is a space monomial curve. Let \(k\) be a field of characteristic \(0\) and \(A = k[x,y,z]\) a polynomial ring. Let \(a\), \(b\), \(c\) be pairwise coprime positive integers and \(P \subset A\) the defining ideal of the curve \((t^a, t^b, t^c)\) in \(k^3\). If \(P\) is not complete intersection, then we know that \[ P = ( x^{s_2 + s_3} - y^{t_1} z^{t_1}, y^{t_1 + t_3} - x^{s_2} z^{u_2}, z^{u_1 + u_2} - x^{s_3} y^{t_3}) \] with positive integers \[ s_2, s_3, t_1, t_3, u_1, u_2 \tag{1} \] The main theorem of this paper gives a necessary and sufficient condition for the symbolic Rees algebra of \(P\) to be Noetherian by enumerating lattice points in a triangle determined by integers (1) provided \(\sqrt{abc} > (u_1 + u_2)c\).