math.AC — Commutative Algebra

A counterexample to the Eisenbud–Mazur conjecture

Contributed by Chung Pang Mok, Yiqi Xu

We construct a height-five prime ideal $P$ in the six-variable formal power series ring $R$ over $\mathbb{Q}$ such that $P^{(2)}\not\subseteq\mathfrak{m}P$, thus providing a prime-ideal counterexample to the Eisenbud--Mazur conjecture (with $\mathbb{Q}$ being the coefficient field). The ideal is the kernel of an explicit quadratic parametrization over a number field of degree $28$ over $\mathbb{Q}$. A witness has order three, with leading term the determinant of a symmetric $3\times3$ matrix. We construct the witness by eliminating critical values to all orders and then straightening the resulting formal curve. Interpolation at the $28$ nodes of an arrangement of eight lines supplies the required lifting maps. A rational linear functional on third and fourth jets certifies nonmembership in $\mathfrak{m}P$. The construction also yields an unmixed radical ideal with $28$ smooth branches over $\mathbb{C}$, and provides a counterexample to the more general form of the Eisenbud--Mazur conjecture stated by More (with $\mathbb{C}$ being the coefficient field). Finally, the prime ideal counterexample descends to the localized polynomial ring $\mathbb{Q}[x_0,\ldots,x_5]_{(x_0,\ldots,x_5)}$, where a polynomial witness with the same fourth jet exists. All finite identities used in the proof admit exact verification.