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A counterexample to the Eisenbud–Mazur conjecture

A mix of human-written and AI-generated textHuman understanding: all partsmath.AC — Commutative Algebramath.AG — Algebraic Geometry

Contributed by Chung Pang Mok, Yiqi Xu ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We construct a height-five prime ideal PP in the six-variable formal power series ring RR over Q\mathbb{Q} such that P(2)⊈mPP^{(2)}\not\subseteq\mathfrak{m}P, thus providing a prime-ideal counterexample to the Eisenbud--Mazur conjecture (with Q\mathbb{Q} being the coefficient field). The ideal is the kernel of an explicit quadratic parametrization over a number field of degree 2828 over Q\mathbb{Q}. A witness has order three, with leading term the determinant of a symmetric 3×33\times3 matrix. We construct the witness by eliminating critical values to all orders and then straightening the resulting formal curve. Interpolation at the 2828 nodes of an arrangement of eight lines supplies the required lifting maps. A rational linear functional on third and fourth jets certifies nonmembership in mP\mathfrak{m}P. The construction also yields an unmixed radical ideal with 2828 smooth branches over C\mathbb{C}, and provides a counterexample to the more general form of the Eisenbud--Mazur conjecture stated by More (with C\mathbb{C} being the coefficient field). Finally, the prime ideal counterexample descends to the localized polynomial ring Q[x0,…,x5](x0,…,x5)\mathbb{Q}[x_0,\ldots,x_5]_{(x_0,\ldots,x_5)}, where a polynomial witness with the same fourth jet exists. All finite identities used in the proof admit exact verification.

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References

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  5. T. Murayama, Uniform bounds on symbolic powers in regular rings , arXiv:2111.06049v5 [math.AC] (https://arxiv.org/abs/2111.06049v5), 2023.arXiv

Version history

  1. v1Submitted by Yiqi XuInitial depositCurrentOct 19, 2026