Set-theoretic complete intersection problem for affine curves in three-dimensional space

Determine whether every affine curve in three-dimensional affine space (equivalently, every irreducible affine curve defined in k[x, y, z]) is set-theoretically a complete intersection; that is, ascertain whether or not there exists a three-dimensional affine curve that fails to be set-theoretically a complete intersection.

Background

The paper constructs explicit families of prime ideals in k[[x, y, z]] with unbounded minimal numbers of generators and stresses that such constructions may shed light on classical questions about set-theoretic complete intersections in embedding dimension three.

The authors point out that understanding the set-theoretic complete intersection property for curves embedded in three-dimensional affine space remains a longstanding open problem and motivate their work as potentially relevant to this question.

References

As said by Moh, obtaining families of prime ideals in R = k[x, y, z] or A = k[x, y, z] which need an arbitrarily large number of generators may lead to help to understand the open problem of whether or not a three-dimensional affine curve is set-theoretically a complete intersection.

— Explicit minimal generating sets of a family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring  (2604.00638 - González et al., 1 Apr 2026) in Section 1. Introduction