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arXiv · 2604.07519

A one-step counterexample to the normalized Nash blowup conjecture

Alvaro Liendo·Ana Julisa Palomino·Gonzalo Rodríguez
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Abstract

We construct an explicit normal singular affine toric variety X of dimension five over an algebraically closed field of characteristic three such that the normalized Nash blowup of X already contains an open affine subset isomorphic to X. Combined with previously known examples, this yields one-step counterexamples in every dimension greater than or equal to five and every characteristic. The characteristic-three case is the most delicate: the previously known counterexample in dimension four requires a two-step iteration of the normalized Nash blowup, and our example demonstrates that in dimension five and higher the minimal number of iterations needed to produce a loop is one.

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Alvaro Liendo, Ana Julisa Palomino, Gonzalo Rodríguez. 2026-04-08. A one-step counterexample to the normalized Nash blowup conjecture. https://arxiv.org/abs/2604.07519

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