A one-step counterexample to the normalized Nash blowup conjecture

TL;DR

This paper presents a specific five-dimensional normal affine toric variety over a field of characteristic three, where the normalized Nash blowup is already an isomorphism, providing a one-step counterexample to the conjecture.

Contribution

It constructs the first explicit one-step counterexample to the normalized Nash blowup conjecture in dimension five and higher, in characteristic three.

Findings

The normalized Nash blowup of the constructed variety contains an open subset isomorphic to the original variety.

Counterexamples exist in all dimensions ≥5 and in all characteristics.

In dimension four, a two-step iteration is necessary, but in higher dimensions, one step suffices.

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.