Symbolic Rees algebras of space monomial primes of degree 5

TL;DR

The paper proves that the symbolic Rees algebra $R_s(\mathfrak{p}_K(5,103,169))$ is not Noetherian over a field $K$ of characteristic $0$, addressing a long-standing open case for space monomial primes of degree $5$. It combines toric geometry and lattice-polygon (Erhart) methods to exhibit a negative curve in $[\mathfrak{p}_K(5,103,169)^{(7)}]_{2065}$, and it develops Huneke's criterion to connect the finite generation of $R_s$ with the existence of a second element $g_2$ satisfying Huneke's condition alongside a fixed negative curve. The authors then derive a contradiction by reduction to characteristic $2$ and a detailed computational check showing $[\mathfrak{p}^{(59)}]_{17407}=0$, which rules out finite generation. This result ties into broader themes around Nagata-type questions and Cox rings, illustrating how geometric and combinatorial techniques can decide finite generation for symbolic Rees algebras in subtle cases.

Abstract

Let K be a field of characteristic 0. Let P_K(5,103,169) be the defining ideal of the space monomial curve {(t^5,t^{103},t^{169})}. In this paper we shall prove that the symbolic Rees algebra R_s(P_K(5,103,169)) is not Noetherian, that is, is not finitely generated over K.

Theorems & Definitions (12)

• Theorem 1.1
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• Proposition 2.6
• Theorem 3.1
• Remark 3.2
• Lemma 3.3