Stabilization of iterated Nash blow-ups along non-divisorial valuations: extending Hironaka's divisorial stabilization theorem

Hironaka (1983) proved that over a field of characteristic zero, every Nash sequence (a sequence of proper birational maps whose pullback of the sheaf of Kähler differentials, modulo torsion, is locally free) stabilizes along the center of any divisorial valuation ring of the function field: past some index the centers are smooth points and the transition maps are local isomorphisms there. Hironaka's theorem is stated only for divisorial valuations; the question below asks whether the analogous stabilization holds for non-divisorial valuations. Valuation-theoretic local uniformization by iterated (normalized) Nash blow-ups is known for maximal-rank monomial valuations on toric varieties and for a large class of valuations on toric surfaces, while it is open for rank-one valuations of rational rank two on toric surfaces, for non-monomial valuations on arbitrary varieties, and in the two remaining characteristic-zero regimes of the Semple–Nash termination conjecture (dimension 2 without normalization, dimension 3 with normalization), where the valuation-stabilization question and the termination conjecture stand or fall together by an elementary domination argument.

Background

Let $k$ be a field and $X \subset \mathbb{A}^n_k$ a reduced, equidimensional variety of dimension $d$ . The Nash blow-up (introduced by Semple and developed by Nobile and Hironaka; also called the Nash–Semple modification) is the proper birational map $\mathcal{N}(X) \to X$ obtained as the Zariski closure of the graph of the Gauss map $x \mapsto T_xX$ , which sends a smooth point to its embedded tangent space in the Grassmannian $\mathrm{Gr}(d,n)$ ; the construction is independent of the embedding. By a universal property due to Nobile, it is the terminal birational map $\tau: Y \to X$ such that the pullback $\tau^*\Omega_X$ of the sheaf $\Omega_X$ of Kähler differentials (the coherent sheaf generated locally by the symbols $df$ for regular functions $f$ , modulo the Leibniz relations), taken modulo its torsion subsheaf, is locally free. Composing with the normalization of $Y$ gives the normalized Nash blow-up.

Hironaka (1983) studied arbitrary Nash sequences: sequences $X_0 = X \xleftarrow{\sigma_1} X_1 \xleftarrow{\sigma_2} X_2 \leftarrow \cdots$ in which each $X_i$ is reduced and equidimensional, each $\sigma_i$ is proper and birational, and each $\sigma_i^*\Omega_{X_{i-1}}$ modulo torsion is locally free. Iterating the (normalized) Nash blow-up produces such a sequence, and every such sequence factors through the Nash blow-up of the previous term, so Nash sequences are exactly the sequences reachable by iterating Nash-type blow-ups.

A valuation ring of $K/k$ , where $K = k(X)$ is the field of rational functions, is a valuation ring $V$ of $K$ containing $k$ ; its center on a model $X_i$ is the unique point $\xi_i$ whose local ring is dominated by $V$ . The valuation $V$ is divisorial if its residue field has transcendence degree $\dim X - 1$ over $k$ ; equivalently, $V$ is the local ring at the generic point of a prime divisor on some normal model of $K$ , so divisorial valuations are exactly those realized geometrically by one equation. Non-divisorial valuations include curve valuations (order of vanishing along a formal, not necessarily algebraic, arc), infinitely singular valuations (rank-one valuations whose value group is not finitely generated, hence not expressible by finite algebraic data), and monomial valuations with irrational or rationally independent weights; the rational rank of a valuation is the dimension of $\mathbb{Q} \otimes_{\mathbb{Z}} \Gamma$ for its value group $\Gamma$ , and it exceeds one precisely for valuations with rationally independent weight vectors.

Hironaka's theorem: over a field of characteristic zero, for any Nash sequence and any divisorial valuation ring $V$ of $K/k$ with centers $\xi_i$ , there is an index $i_0$ such that for all $i \ge i_0$ the variety $X_i$ is smooth at $\xi_i$ and $\sigma_{i+1}$ is a local isomorphism at $\xi_{i+1}$ . Hironaka's 1983 paper states no result and poses no question about non-divisorial valuations.

The surrounding program: Semple (1954) proposed iterating the modification to resolve singularities. Nobile (1975) proved that in characteristic zero the Nash blow-up of $X$ is an isomorphism exactly when $X$ is smooth, and deduced that iterated Nash blow-ups resolve all characteristic-zero curves. Spivakovsky (1990), building on Hironaka's theorem and on the normalized variant, proved that iterated normalized Nash blow-ups resolve all complex surface singularities. Castillo–Duarte–Leyton-Álvarez–Liendo (2026) showed by explicit normal toric examples that in every dimension at least four, over an algebraically closed field of any characteristic, these iterations fail to resolve singularities: in characteristic zero the Nash blow-up and the normalized Nash blow-up of their example each contain an open affine subset isomorphic to the example itself, so both processes loop forever, and in prime characteristic the normalized iteration loops; a 2025 companion paper by the same authors shows that already in dimension three, in characteristic zero, the second iteration of the (non-normalized) Nash blow-up of a non-normal toric variety contains the original variety. In parallel, González Pérez–Teissier and Duarte developed the local uniformization face of the question — the classical problem, going back to Zariski, of finding a model on which a given valuation has a smooth center — for monomial valuations on toric varieties, where the Nash blow-up computes the blow-up of the logarithmic Jacobian ideal (the ideal generated by the maximal minors of the matrix of logarithmic derivatives along the dense torus) and valuations of the toric function field correspond to points of the Zariski–Riemann space, the direct-limit space of all valuation rings of $K/k$ ordered by domination. Whether iterated (normalized) Nash blow-ups achieve local uniformization along a given valuation is precisely the stabilization question, restricted to the canonical process.

Problem Statement

Hironaka's stabilization theorem for Nash sequences is stated for divisorial valuation rings of the function field. Determine whether, and for exactly which classes of valuations and processes, the analogous stabilization holds for non-divisorial valuations.

Precisely: let $k$ be a field of characteristic zero, $X$ a reduced equidimensional $k$ -variety with function field $K$ , and let $X_0 = X \xleftarrow{\sigma_1} X_1 \xleftarrow{\sigma_2} \cdots$ be the sequence obtained by iterating the Nash blow-up of $X$ (respectively, the Nash blow-up followed by normalization). For which valuations $V$ of $K/k$ that are not divisorial does there exist an index $i_0$ such that for all $i \ge i_0$ the center $\xi_i$ of $V$ on $X_i$ is a smooth point of $X_i$ and $\sigma_{i+1}$ is a local isomorphism at $\xi_{i+1}$ ?

The restriction to the canonical iterated process is necessary. Taken verbatim over arbitrary Nash sequences — Hironaka's theorem quantifies over all sequences of proper birational maps whose pullback of the sheaf of Kähler differentials, modulo torsion, is locally free — the extension to non-divisorial valuations is false for an elementary reason: on a smooth surface, blowing up an infinite chain of infinitely near points along a formal arc yields a Nash sequence whose associated curve valuation is non-divisorial and whose step maps are never local isomorphisms at its centers. The meaningful question therefore concerns the canonical iterated (normalized) Nash process and specified classes of valuations.

Concrete open instances of this question, none of which is settled in the cited literature; here and below a regime means the part of the corresponding dimension/process/characteristic bucket that the cited sources leave open, so a valuation class whose stabilization is already established in the cited sources lies outside the regime: (1) dimension two, non-normalized iteration, characteristic zero — the last open case of the Semple–Nash conjecture in that setting, where stabilization along every non-divisorial valuation is equivalent to termination of the iteration: termination makes every center smooth, while failure of termination yields, by the standard Zariski–Riemann domination argument, a valuation whose center is singular on every iterate, necessarily non-divisorial by Hironaka's theorem; (2) dimension three, normalized iteration, characteristic zero — likewise open; (3) the rank-one valuations of rational rank two with irrational coordinate ratio on toric surfaces, left open by Duarte's Theorem 0.1; (4) non-monomial valuations (curve valuations, infinitely singular valuations, formal-arc valuations) on arbitrary varieties, together with the version for all preorders of the Zariski–Riemann space left open in Remark 14.6 of González Pérez–Teissier.

The question is settled by a proof of stabilization for a stated class of non-divisorial valuations under a stated process (a class that is left open by the cited sources, is taken over a variety that is singular at the centers of those valuations, and equivalently lies in one of the regimes listed above), by an explicit variety, an explicitly iterated (normalized) Nash process, and a non-divisorial valuation witnessing failure of stabilization in one of those regimes, or by determining, for the regimes listed above, which of them admit stabilization and which do not. The four regimes are independent sub-questions: settling stabilization in any one of them does not settle the others. The question extends the exact conclusion of Hironaka's 1983 theorem beyond the divisorial hypothesis under which that theorem is proved, in the setting of the canonical iterated process and specified valuation classes.

References

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Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez and Alvaro Liendo, Non-normalized Nash blowup fails to resolve singularities in dimension three, arXiv:2511.01772 [math.AG] (2025), https://arxiv.org/abs/2511.01772.
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