On the symbolic powers of defining ideals of monomial curves associated to generalized arithmetic sequences (Q6594896)

Let \(R=k[t^a, t^{sa+d}, t^{sa+2d}, \ldots, t^{sa+nd}]\), where \(k\) is a field and \(a,s,n,d\) are positive integers with \(n \geq 2\) and \(\operatorname{gcd}(a,d)=1\). Let \(P\) be the defining ideal of \(R\), that is, the kernel of the \(k\)-algebra homomorphism \(k[x_0, x_1, \ldots, x_n]\to R\) that maps \(x_0\) to \(t^a\) and \(x_i\) to \(t^{sa+id}\) for \(2 \leq i \leq n\). In this context, the paper studies the question of when the \(i\)-th symbolic power \(P^{(i)}\) coincides with the power \(P^i\). The following are established:\N\N\begin{itemize}\N\item[1.] If \(n=2\) and \(a\) is even, then \(P^{(i)}=P^i\) for every \(i>0\).\N\item[2.] If \(n=2\) and \(a\) is odd, then \(P^{(i)}\neq P^i\) for every \(i>0\).\N\item[3.] If \(n=3\) and \(a \equiv 2\) (mod 3), then \(P^{(2)}= P^2\) and \(P^{(3)}\neq P^3\).\N\item[4.] If \(n=3\) and \(a \not\equiv 2\) (mod 3), then \(P^{(2)}\neq P^2\).\N\item[5.] If \(n \geq 4\), then \(P^{(2)}\neq P^2\).\N\end{itemize}\N\NUnder the additional assumption \(\operatorname{char} k =0\), the author also considers the noetherian property of the symbolic Rees algebra \(\mathcal{R}_s(P)=\oplus_{i \geq 0} P^{(i)}\). It is noted that \(\mathcal{R}_s(P)\) is noetherian in the following cases: (1) \(n=2\); (2) \(a=n+1\); and (3) \(n=3\) and \(a=5\).