A counterexample to the Eisenbud–Mazur conjecture
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…
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