Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves

Abstract

In this paper, we study finite generation of symbolic Rees rings of the defining ideal $\mathfrak{p}$ of the space monomial curve $(t^a,t^b,t^c)$ for pairwise coprime integers $a$, $b$, $c$. Suppose that the base field is of characteristic $0$, and the ideal $\mathfrak{p}$ is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element $\xi$ of the minimal degree in $\mathfrak{p}$ is a negative curve, we determine the minimal degree of an element $\eta$ such that the pair $\{\xi ,\eta \}$ satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring ${\mathcal{R}}_s(\mathfrak{p})$ is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of $\mathfrak{p}$ in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of $\mathfrak{p}$ in which the homogeneous element of the minimal degree in ${\mathfrak{p}}^{(2)}$ is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.