Symbolic Rees Rings and Nagata’s Conjecture

Definitions and mathematical statement

Let $a,b,c\in{\mathbb{Z}}_{>0}$ satisfy $\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1$. To specify the algebraic notation, fix a field $k$ and put $A=k[x,y,z]$. The choice and characteristic of $k$ are not specified in the supplied source record; the following construction is therefore relative to this auxiliary choice. Define \[ \varphi_{a,b,c}:A\longrightarrow k[t],\qquad x\longmapsto t^a,\quad y\longmapsto t^b,\quad z\longmapsto t^c, \qquad P=P(a,b,c):=\ker(\varphi_{a,b,c}). \] Thus $P$ is the prime ideal defining the affine monomial curve parametrized by $(t^a,t^b,t^c)$. Set $P^{(0)}=A$, and for every integer $m\geq1$ define the symbolic power \[ P^{(m)}:=P^mA_P\cap A. \] With $T$ an indeterminate recording the grading, its symbolic Rees ring is \[ R_s(P):=\bigoplus_{m\geq0}P^{(m)}T^m\subseteq A[T]. \] Problem. Characterize the pairwise coprime positive triples $(a,b,c)$ for which $R_s(P(a,b,c))$ is finitely generated as an $A$-algebra, equivalently Noetherian, with the necessary hypotheses on $k$ made explicit. Determine the relationship between this finite-generation problem and Nagata's conjecture for general points in $\mathbb P^2$.

Unresolved qualifications. The source does not specify the ground field, its characteristic, or whether the intended characterization is uniform over fields. It also does not state a precise version of Nagata's conjecture, the number or multiplicities of the points, the meaning of generality, or a proposed implication or equivalence. Consequently, the requested relationship cannot be expressed as a uniquely determined logical assertion from the supplied record alone. The positive-integer hypotheses permit an exponent equal to $1$; no exclusion of such cases is supplied.

Short English statement

For pairwise coprime positive integers $a,b,c$, characterize when the symbolic Rees ring of the monomial curve $(t^a,t^b,t^c)$ is Noetherian, and explain its relationship to Nagata's conjecture for general plane points. The ground-field assumptions and the precise conjectural relationship remain unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001241$, exact source problem code catalogue entry, titled "Symbolic Rees Rings and Nagata's Conjecture," in the Oberwolfach Reports Open Problems collection. The record assigns the year 2009 and supplies no named proposer.
• Kommutative Algebra (2009), Oberwolfach Reports, Report 22, pp. 1157--1224, DOI 10.4171/owr/2009/22. Supplied official report reference: https://doi.org/10.4171/owr/2009/22.
• Supplied literature reference labeled "Kurano-Matsuoka on space monomial curves": https://arxiv.org/abs/0801.3896.
• Supplied literature reference labeled "Kurano-Nishida on infinitely generated examples": https://arxiv.org/abs/1705.09865.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes its connection to Nagata-type negative-curve questions, with later papers resolving families and examples but not the general problem. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification description is supplied metadata and has not been independently confirmed for this draft.