Infinitely generated symbolic Rees rings of space monomial curves having negative curves

In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal ${\frak p}$ 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 above ideal ${\frak p}$ is minimally generated by three polynomials. Under the assumption that the homogeneous element $\xi$ of the minimal degree in ${\frak p}$ is the 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 ${\cal R}_s({\frak p})$ is Notherian using computers. We give a necessary and sufficient conditions for finite generation of the symbolic Rees ring of ${\frak p}$ under some assumptions. We give an example of an infinitely generated symbolic Rees ring of ${\frak p}$ in which the homogeneous element of the minimal degree in ${\frak p}^{(2)}$ is the negative curve. We give a simple proof to (generalized) Huneke's criterion.