In this paper, we study finite generation of symbolic Rees rings of the defining ideal of the space monomial curve for pairwise coprime integers , , . Suppose that the base field is of characteristic , and the ideal is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element of the minimal degree in is a negative curve, we determine the minimal degree of an element such that the pair 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 is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of in which the homogeneous element of the minimal degree in is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.
"Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves." Michigan Math. J. 68 (2) 409 - 445, June 2019. https://doi.org/10.1307/mmj/1557475399