Nash Valuations on Terminal 3-Fold Singularities

• The paper establishes that every terminal valuation from a minimal model corresponds to a Nash valuation via a surjective Nash map.
• It employs tools such as Reguera’s curve-selection lemma, Hurwitz-type Jacobian calculations, and the negativity lemma to substantiate its claims.
• Examples involving terminal toric and cAx/2 singularities illustrate both the classification of Nash valuations and the map's limitations in certain cases.

A Nash valuation on a 3-fold terminal singularity is a divisorial valuation defined by a prime exceptional divisor over the singularity that arises as the valuation attached to a maximal family of arcs through the singular locus, via the Nash map. Recent advances have established the precise relationship between terminal valuations—those divisorial valuations realized on some minimal model as exceptional divisors with positive discrepancy—and Nash valuations in dimension three. This link provides a geometric bridge between the minimal model program (MMP) and the structure of arc spaces.

1. Terminal Valuations and the Minimal Model Program

Let $X$ be a normal, projective variety of characteristic zero with $\dim X = 3$. A minimal model of $X$ is a projective birational morphism $f: Y \to X$ such that $Y$ is normal, has only terminal singularities, and $K_Y$ is $f$-nef. The canonical divisor on $Y$ decomposes as

$K_Y = f^*K_X + \sum_i a(E_i,X) E_i,$

where $E_i$ are $f$-exceptional divisors and $a(E_i,X)>0$ are discrepancies. A divisorial valuation $v=\operatorname{ord}_E$ is called terminal if $E$ is a prime exceptional divisor on some minimal model and $a(E,X)>0$ (Fernex et al., 2014).

2. The Nash Map, Arc Spaces, and Nash Valuations

Arc spaces $X_\infty$ parametrize formal arcs $\alpha: \operatorname{Spec} k[[t]] \rightarrow X$. The Nash map assigns to each irreducible component $Z \subset X_\infty$ of arcs centered in $X_{\mathrm{sing}}$ the valuation defined by the generic arc. Such valuations are called Nash valuations.

For 3-folds $X$ with only terminal singularities, every terminal valuation is a Nash valuation: to every prime exceptional divisor $E$ on a minimal model, there corresponds an irreducible component $N_E \subset X_\infty$ whose generic arc has valuation $\operatorname{ord}_E$ (Fernex et al., 2014). Thus, the Nash map is surjective onto terminal valuations in dimension three.

3. Main Theorems, Proof Techniques, and Key Lemmas

The crucial result is that for any $X$ as above and any terminal valuation $v = \operatorname{ord}_E$, there exists a Nash component whose associated valuation is $v$. The argument utilizes Reguera’s curve-selection lemma: assuming for contradiction that $N_E$ is not maximal leads to constructing a wedge that yields impossible inequalities among discrepancies, using the Hurwitz formula and the negativity lemma (Fernex et al., 2014). The contradiction establishes the surjectivity of the Nash map onto terminal valuations.

Key technical ingredients include:

• Negativity lemma: For an $h$-exceptional $\mathbb Q$-divisor $D$ with nonnegative intersection with all $h$-exceptional curves, one has $D \leq 0$.
• Hurwitz-type Jacobian calculation: Relates coefficients of divisors in the relative canonical class to ramification.

4. Structure and Examples of 3-Fold Terminal Singularities

Threefold terminal singularities are classified, up to analytic isomorphism, as either Gorenstein $cDV$ points (index 1) or hyperquotients of the form $$(\phi=0) \subset \mathbb C^4/\frac{1}{m}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$$. Notable examples include:

• Terminal quotient singularities: E.g., $X=\mathbb{A}^3/\mu_r(1,a,b)$, where minimal models correspond to crepant subdivisions, and exceptional divisors correspond to Nash valuations (Fernex et al., 2014).
• $cAx/2$ singularities: For $$X= \{x^2 + y^2 + f(z,u) = 0\} \subset \mathbb C^4/\frac{1}{2}(0,1,1,1)$$ with $f(z,u) \in (z,u)^4$, every exceptional divisor computing the minimal discrepancy $1/2$ defines a Nash valuation (Lin, 20 Dec 2025).

Table: Examples and Nash Map Surjectivity

Singularities	Minimal Discrepancy	Nash Map Surjectivity
Terminal toric 3-fold	$a(E,X)\leq 1$	Surjective onto minimal-discrepancy E
$cAx/2$-points	$a(E,X)=1/2$	Surjective for divisors with $a=1/2$
Gorenstein $cA$	$a(E,X)=1$	Surjective

5. Classification Results and Counterexamples

For $cA/r$ threefolds, the classification of Nash valuations and essential valuations depends on the Gorenstein and $\mathbb Q$-factorial nature of the singularity:

• In the Gorenstein case ($r=1$), all exceptional divisors with $a(E,X)\leq 1$ correspond to Nash valuations (Chen, 2019).
• For $r>1$, every essential divisor satisfies $a(E,X) \leq 2$, and the Nash map is surjective under certain numerical conditions on $f(z,u)$.
• Explicit counterexamples show surjectivity fails beyond these settings; i.e., there exist essential divisors that do not arise as Nash valuations (Chen, 2019).

Further, de Fernex constructed 3-fold hypersurface singularities where the Nash map is not surjective: two divisorial valuations arise in the resolution process, both essential, but only one corresponds to a Nash valuation (Fernex, 2012).

6. Conjectures and Open Problems

Several conjectures aim to generalize the observed correspondence between minimal discrepancy and Nash valuations:

• Conjecture A: Every exceptional divisor with $a(E,X)\leq 1$ induces a Nash valuation.
• Conjecture B: Every exceptional divisor with minimal discrepancy $a(E,X)=1/m$ induces a Nash valuation (Lin, 20 Dec 2025).

Partial confirmations exist for toric, $cA/m$, $cAx/2$, and certain $cD$, $cE$ points. The broader classification for all 3-fold terminal singularities—particularly for the families $cAx/4$, $cD/2$, $cD/3$, $cE/2$—remains open (Lin, 20 Dec 2025), and the complete structure of the image of the Nash map in higher dimensions is yet to be resolved.

7. Birational, Toric, and Analytic Perspectives

In toric cases, all minimal discrepancy divisors yield Nash valuations, and explicit combinatorial descriptions exist. The image of the Nash map coincides with minimal-valuation loci in toric 3-folds (Fernex et al., 2014, Lin, 20 Dec 2025). The distinction between algebraic and analytic categories is significant; some divisors are essential algebraically but not analytically, as proven for specific examples in (Fernex, 2012).

The study of Nash valuations on 3-fold terminal singularities thus intertwines birational geometry, arc space theory, and explicit singularity analysis, with the surjectivity of the Nash map fully understood only in selected cases and several intriguing open questions remaining.