Normalized Nash Blowup for Toric Varieties

• Normalized Nash blowup is a refined modification that normalizes the Nash blowup by closing the Gauss map’s graph, capturing tangent spaces of smooth loci.
• It employs combinatorial tools like monomial ideals and Newton polyhedra to derive explicit toric constructions and resolve singularities effectively.
• The method yields finite resolutions in toric surfaces and specific threefolds, though it faces limitations in dimension four and higher.

The normalized Nash blowup is a modification of an algebraic variety that refines the classical Nash blowup by taking the normalization of the Zariski closure of the graph of the Gauss map from the smooth locus of the variety to the Grassmannian of tangent spaces. For toric varieties, and particularly in positive characteristic, the normalized Nash blowup acquires a compelling combinatorial description via monomial ideals and Newton polyhedra, central to advances and counterexamples in resolution of singularities.

1. Definition and Core Construction

Given an equidimensional variety $X \subset K^n$ of dimension $d$ over an algebraically closed field $K$ (arbitrary characteristic), the smooth locus $X \setminus \operatorname{Sing}(X)$ admits the Gauss map

$$G : X \setminus \operatorname{Sing}(X) \to \operatorname{Gr}(d, n)$$

sending each $x$ to its tangent space $T_x X$. The Nash blowup $X^*$ is the Zariski closure in $X \times \operatorname{Gr}(d, n)$ of the graph of $G$, with projection $\pi: X^* \to X$ a proper, birational morphism, isomorphic over the smooth locus. The normalized Nash blowup is the normalization $\widetilde{X}$ of $X^*$, with the composite map $\widetilde{X} \to X^* \to X$ called the normalized Nash blowup.

For an affine toric variety $X = \operatorname{Spec} K[T]$, with $T$ a normal subsemigroup of $\mathbb{Z}^d$, $X$ is defined by a strongly convex rational polyhedral cone $\sigma \subset \mathbb{R}^d$, with $T = \sigma^\vee \cap \mathbb{Z}^d$. The Nash and normalized Nash blowups admit a combinatorial algorithm based on the Hilbert basis of $T$ and associated polyhedra. In positive characteristic, careful attention to the behavior of determinants modulo $p$ is essential for correct construction of the blowup and its normal fan (Duarte et al., 2022, Castillo et al., 16 Jan 2025, Castillo et al., 22 Nov 2025).

2. Combinatorial Description: Logarithmic Jacobian and Newton Polyhedron

The core combinatorial tool is the logarithmic Jacobian ideal. For $T$ generated minimally by $\{\nu_1, \ldots, \nu_n\}$, the log-Jacobian ideal in characteristic 0 is

$$J_0 = \left( \det(\nu_{i_1}, \ldots, \nu_{i_d}) \;\bigg|\; \det(\nu_{i_1}, \ldots, \nu_{i_d}) \ne 0 \right) \subset K[x_1, \ldots, x_n].$$

In characteristic $p>0$,

$$J_p = \left( \det(\nu_{i_1}, \ldots, \nu_{i_d}) \;\bigg|\; \det(\nu_{i_1}, \ldots, \nu_{i_d}) \not\equiv 0 \pmod p \right) \subset K[x_1, \ldots, x_n].$$

The Nash blowup, and thus its normalization, is isomorphic to the blowup along $J_p$. The Newton polyhedron $New(J_p)$ is defined as

$$\operatorname{Conv}\left\{ \nu_{i_1} + \cdots + \nu_{i_d} + \mathbb{R}_{\geq 0}^d \mid \det(\nu_{i_1}, \ldots, \nu_{i_d}) \not\equiv 0 \pmod p \right\}$$

The normal fan of $New(J_p)$ defines the toric modification corresponding to the normalized Nash blowup (Duarte et al., 2022, Castillo et al., 16 Jan 2025).

3. Resolution Properties and Known Families

3.1 Toric Surfaces

A central result is that for normal toric surfaces (dimension 2), normalized Nash blowups always yield a resolution of singularities after finitely many steps over any algebraically closed base field of arbitrary characteristic:

$$X_T = X^0 \xleftarrow{} \widetilde{X}^1 \xleftarrow{} \widetilde{X}^2 \xleftarrow{} \cdots \xleftarrow{} \widetilde{X}^N$$

produces a smooth variety for some $N$ (Duarte et al., 2022, Castillo et al., 16 Jan 2025). The combinatorics and induced fan subdivisions coincide with those in characteristic zero.

3.2 Characteristic-Free Scenarios

For specific classes of cones, the Newton polyhedron $N_p(\sigma)$ and the corresponding fan do not depend on the characteristic. Sufficient combinatorial criteria include G-stable cones (where the Hilbert basis lies on the compact faces and behaves well under subcone restriction) and cones from G-flat smooth polytopes. In such cases, the normalized Nash blowup is smooth over any characteristic (Castillo et al., 16 Jan 2025). Examples include all normal affine toric surfaces, 2-generic determinantal varieties, and affine toric varieties from smooth G-flat polytopes.

3.3 Higher Nash Blowups of Toric Surfaces

The normalization of the higher Nash blowup of order $n$ for the toric surface singularity $A_n$ factors through its minimal resolution. Explicit combinatorial constructions, including Gröbner fans of powers of monomial ideals, refine the standard toric resolution fan, but do not introduce new toric exceptional divisors beyond the minimal resolution (Chávez-Martínez, 2021).

4. Limitations, Dependence on Characteristic, and Counterexamples

4.1 Dimension Three

In dimension 3, explicit families show that the normalized Nash blowup can depend on the characteristic, with the Newton polyhedron $N_p(\sigma)$ varying even for fixed combinatorics. For certain three-dimensional cones, the set of vertices of $New(J_p)$ differs between characteristics $p=0$ and $p=2$, leading to different singularity behavior after the first blowup (Duarte et al., 2022).

4.2 Dimensions Four and Higher

In every dimension $d \geq 4$, there exist explicit normal affine toric varieties such that no finite sequence of (normalized) Nash blowups resolves singularities. In these constructions, after one (or more) steps, an open affine chart is isomorphic to the original singularity, leading to infinite loops and failure of the process (Castillo et al., 29 Sep 2024, Castillo et al., 22 Nov 2025). This holds for all characteristics, including characteristic zero, and covers important classes such as toric hypersurfaces, cyclic quotient singularities, and certain $\mathbb{Q}$-factorial Gorenstein singularities.

4.3 Failure on Non-Surface Toric Varieties

While the conjecture of resolving all singularities via normalized Nash blowup holds for toric surfaces and many three-dimensional cases (with extensive computational evidence), it is now settled by explicit counterexamples that the conjecture fails in dimensions four and above, both for Nash and normalized Nash blowups (Castillo et al., 29 Sep 2024, Castillo et al., 22 Nov 2025).

5. Positive Results, Computational Evidence, and Open Cases

5.1 Large-scale Computations

Recent implementations based on the combinatorial algorithm—computing Hilbert bases, forming Newton polyhedra, and constructing normal fans—have provided broad positive evidence for the resolution of singularities by iterated normalized Nash blowups in dimension 2 (toric surfaces) and for threefolds in characteristic zero. No cycles or fails have been found among $50,000$ surface cases and $2.5$ million threefold cases, respectively (Castillo et al., 22 Nov 2025).

5.2 Remaining Open Cases

- Ordinary Nash blowup conjecture (resolution in all dimensions via Nash blowups) remains open only for surfaces in characteristic zero. - Normalized Nash blowup conjecture (resolution via normalization at each step) remains open for threefolds in characteristic zero and all toric singularities not of product type in dimensions 2 and 3 in positive characteristic. - No isolated (non-toric) singularity is yet known to resist normalized Nash iteration in dimension 2 or 3.

A major open problem is to find a monotonic invariant or toric measure that guarantees termination or to close the status in the remaining low-dimensional cases.

6. Explicit Examples and Families

Family/Class	Behavior under Iterated Normalized Nash Blowup	Reference
Normal toric surfaces (all char.)	Always resolved in finitely many steps	(Duarte et al., 2022, Castillo et al., 16 Jan 2025)
Higher Nash blowups of $A_n$	Normalization factors through minimal resolution	(Chávez-Martínez, 2021)
2-generic determinantal varieties	Resolved in single blowup, all characteristics	(Castillo et al., 16 Jan 2025)
Cones from smooth G-flat polytopes	Resolved in one normalized Nash blowup	(Castillo et al., 16 Jan 2025)
Toric varieties, $\dim \geq 4$	Counterexamples: fail to resolve, with cycles	(Castillo et al., 29 Sep 2024, Castillo et al., 22 Nov 2025)
Certain 3-folds (char dep.)	Newton polyhedron varies with $p$, may not resolve	(Duarte et al., 2022)

Detailed combinatorial constructions for these families illustrate the precise way in which the normalized Nash blowup interacts with toric combinatorics, polyhedral and semigroup data, and characteristic.

7. Significance and Current Directions

The normalized Nash blowup provides a canonical, combinatorial approach to singularity resolution for broad classes of varieties; in particular, it furnishes a systematic resolution mechanism for all toric surfaces, determinantal varieties, and new families arising from G-stable cones and G-flat polytopes independent of characteristic (Duarte et al., 2022, Castillo et al., 16 Jan 2025). The theoretical limits of the method—most notably its failure in dimension $\geq 4$—are now precisely delineated by explicit examples (Castillo et al., 29 Sep 2024, Castillo et al., 22 Nov 2025). The remaining low-dimensional cases constitute a major focus for current research, both in algorithmic and theoretical directions. The normalized Nash blowup remains a unique bridge between resolution problems, toric geometry, combinatorics, and computational algebraic geometry.