Nash Problem in Singularity Theory

• Nash problem is a central question in singularity theory linking arc families in singular loci with essential divisors on resolved varieties.
• The approach examines whether the injective Nash map is surjective, demonstrating bijectivity for surface singularities and failures in higher dimensions.
• Key methods include the use of arc spaces, wedges, and topological arguments to analyze adjacencies and combinatorial resolution graphs.

The Nash problem is a question in singularity theory that relates the geometry of resolutions of singularities to the structure of arc spaces. For a complex algebraic variety $X$, one considers the space of arcs $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$ and, more specifically, the arcs centered in the singular locus. Nash proved that these arcs break into finitely many irreducible families and defined an injective map from those families to essential divisors over $X$. The problem asks whether this map is surjective. For surface singularities, Nash conjectured that it is bijective; this is true in characteristic $0$, while in higher dimension surjectivity fails in general (Bobadilla et al., 2018, Bobadilla et al., 2011).

1. Arc spaces, essential divisors, and Nash’s map

Let $X$ be a complex algebraic variety, with singular locus $\Sing(X)$. The space of arcs is

$\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$

and the subspace of arcs centered in the singular locus is

$\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$

Nash proved that $\mathcal{L}_{\Sing(X)}(X)$ has finitely many irreducible components, often called families of arcs through the singularity (Bobadilla et al., 2018).

Choose a resolution of singularities

$\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$

An irreducible component $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$0 of the exceptional locus is called essential if, for every other resolution, its birational transform remains an irreducible component of the exceptional set. By Hironaka, there are only finitely many essential components (Bobadilla et al., 2018).

Nash defined a map

$\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$1

sending a family $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$2 to the unique essential divisor $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$3 such that a general arc $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$4 lifts to $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$5 with its origin on $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$6. Nash showed that $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$7 is injective (Bobadilla et al., 2018).

For a normal surface germ $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$8, this can be expressed more concretely. If

$\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$9

is the minimal resolution, then every arc $X$0 on $X$1 lifts uniquely to an arc $X$2 on $X$3, and one defines

$X$4

Its Zariski closure $X$5 is irreducible, and the Nash map is the assignment

$X$6

In the surface case, Nash’s conjecture states that this map is bijective (Pereira, 2010).

2. Adjacencies, wedges, and equivalent formulations

For surfaces, bijectivity of the Nash map is equivalent to the absence of nontrivial inclusions

$X$7

Such an inclusion is called an adjacency. The wedge-theoretic formulation of the problem converts the existence of an adjacency into the existence of a 1-parameter family of arcs (Bobadilla, 2010).

A wedge is a map

$X$8

over a field extension $X$9, with special arc $0$0 and generic arc $0$1 for $0$2. In the complex-analytic setting one works with convergent wedges

$0$3

For essential components $0$4, one says that $0$5 realizes an adjacency $0$6 if the special arc lifts transversely to $0$7 and the generic arc lifts into $0$8 (Bobadilla, 2010).

A refined wedge criterion states that, for a normal surface singularity over an uncountable algebraically closed field of characteristic $0$9, the following are equivalent: adjacency $X$0; existence of a wedge over the base field with special arc transverse to $X$1 and generic arc in $X$2 for any proper closed subset $X$3; and, when $X$4, existence of convergent wedges with prescribed transverse special arc (Bobadilla, 2010). In the formulation used in the proof for surfaces, an essential divisor $X$5 lies in the image of the Nash map if and only if there is no formal or convergent wedge whose special arc lifts transversely to $X$6 but whose generic arc lifts to some other $X$7 (Bobadilla et al., 2011).

The same paper gives an equivalent reformulation in terms of finite morphisms of normal surface germs realizing adjacencies. As a corollary, the image of the Nash map for normal surface singularities is characterized by the combinatorics of a resolution, or equivalently by the topology of the abstract link in the complex-analytic case (Bobadilla, 2010).

3. The solution for surfaces

For algebraic surfaces over an algebraically closed field of characteristic $X$8, the Nash map is bijective: $X$9 This is the theorem of Fernández de Bobadilla and Pe Pereira (Bobadilla et al., 2011).

The proof reduces first to the normal, complex-analytic case. If $\Sing(X)$0 is not normal, one normalizes and resolves the normalization, using a lifting criterion of Reguera via wedges. In the normal case, surjectivity is proved by contradiction: assuming an essential divisor $\Sing(X)$1 is not in the image, one obtains a convergent wedge realizing an adjacency from some $\Sing(X)$2 to $\Sing(X)$3 (Bobadilla et al., 2011).

After passing to a sufficiently small convergent representative

$\Sing(X)$4

one studies the total transform of the image of the wedge in $\Sing(X)$5: $\Sing(X)$6 with central fiber

$\Sing(X)$7

Intersecting each fiber with the exceptional components gives a linear system

$\Sing(X)$8

where $\Sing(X)$9 is the negative-definite intersection matrix and $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$0. A sign lemma for $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$1 yields constraints such as $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$2 and $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$3 (Bobadilla et al., 2011).

The decisive contradiction is topological. For small $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$4, the normalization of the curve $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$5 is a topological disc, so its Euler characteristic is $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$6. On the other hand, by decomposing $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$7 into tubular neighborhoods of the $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$8 and small Milnor balls around the crossing points of $\mathcal{L}(X)=\{\gamma:\Spec\C[[t]]\to X\},$9, one obtains Euler-characteristic estimates that imply

$\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$0

and in fact

$\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$1

Minimality of the resolution forces each summand to be $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$2, with at least one strictly negative. This contradicts the fact that the normalization is a disc, so no such wedge exists, and every essential divisor lies in the image of the Nash map (Bobadilla et al., 2011).

This uniform argument subsumes earlier case-by-case analyses. Rational double points $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$3 and cusp singularities are recovered within the same framework (Bobadilla et al., 2011).

4. Quotient surface singularities and the ADE case

Before the general surface theorem, Pe Pereira proved bijectivity for all two-dimensional quotient singularities: $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$4 In particular, this covers the Du Val singularities $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$5, including the icosahedral singularity $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$6 (Pereira, 2010).

If $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$7 is a quotient singularity, then $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$8 is rational, has a unique minimal resolution, and its dual graph is one of the ADE diagrams. For $\mathcal{L}_{\Sing(X)}(X)=\{\gamma\in\mathcal{L}(X)\mid \gamma(0)\in\Sing(X)\}.$9, the graph is the $\mathcal{L}_{\Sing(X)}(X)$0 Dynkin diagram and each vertex has self-intersection $\mathcal{L}_{\Sing(X)}(X)$1. The essential divisors $\mathcal{L}_{\Sing(X)}(X)$2 correspond one-to-one with the vertices of this graph (Pereira, 2010).

The proof strategy is organized around adjacencies. Most candidate adjacencies are ruled out by intersection-multiplicity estimates after pulling a wedge back to a family of plane curves $\mathcal{L}_{\Sing(X)}(X)$3 via the covering $\mathcal{L}_{\Sing(X)}(X)$4. For carefully chosen test lines $\mathcal{L}_{\Sing(X)}(X)$5, one compares

$\mathcal{L}_{\Sing(X)}(X)$6

For almost all pairs $\mathcal{L}_{\Sing(X)}(X)$7, this numerical inequality fails, excluding the adjacency (Pereira, 2010).

The remaining deep adjacencies involve wedges whose lifted plane family acquires extra branches, called returns. These are analyzed through the Riemann-surface normalizations of $\mathcal{L}_{\Sing(X)}(X)$8, producing topological contradictions when boundary components behave incompatibly unless certain returns occur. In the truly exceptional cases, including certain $\mathcal{L}_{\Sing(X)}(X)$9 and $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$0 adjacencies, one studies the image curve in the miniversal base of the special arc’s germ and shows that any such 1-parameter deformation cannot remain $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$1-constant. A stratification-codimension count then excludes the required wedge (Pereira, 2010).

This gives the first uniform proof that for every two-dimensional quotient singularity the Nash correspondence between arc-families and essential divisors is a bijection, thereby settling the ADE case in particular (Pereira, 2010).

5. Higher dimensions, terminal valuations, and counterexamples

In higher dimension, the original Nash map is no longer surjective in general. Counterexamples are known in every dimension $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$2 (Fernex, 2012). The three-dimensional examples of de Fernex show that one can arrange a two-step blow-up so that an essential divisor does not produce a new irreducible family of arcs: all contact-one arcs on the second exceptional divisor can be deformed into arcs meeting the first exceptional divisor (Fernex, 2012).

One formulation of the higher-dimensional structure is

$\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$3

A minimal model over $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$4 is a projective birational morphism

$\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$5

with $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$6 normal, terminal, and $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$7 $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$8-nef. A terminal valuation is a divisorial valuation $\pi:(\widetilde X,E)\to (X,\Sing(X)), \qquad E=\bigcup_{i=0}^r E_i.$9 defined by a prime exceptional divisor $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$00 on such a minimal model. De Fernex and Docampo proved that every terminal valuation over $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$01 is a Nash valuation (Fernex et al., 2014).

Their proof again proceeds by contradiction using wedges. If a terminal valuation were not Nash, Reguera’s curve-selection lemma would produce a wedge whose special arc meets $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$02 transversely but whose generic arc lies in a larger component of arcs through the singular locus. After resolving indeterminacies of the induced rational map to the minimal model, one compares relative canonical divisors and uses nefness together with negativity to obtain an impossible chain of inequalities, contradicting transversality (Fernex et al., 2014).

In dimension two, the minimal model is the minimal resolution, and terminal valuations coincide with essential valuations. Thus the terminal-valuation theorem recovers the bijectivity of the Nash map for surfaces (Fernex et al., 2014). In dimension $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$03, however, terminal, Nash, and essential valuations need not coincide. The survey of Fernández de Bobadilla and Pe Pereira records that the original Nash map can fail to be surjective in dimension $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$04, and that de Fernex–Docampo identify terminal valuations as the correct higher-dimensional substitute for the surface notion of essential divisors only in a partial sense: every terminal valuation is Nash, but not every essential valuation is terminal or Nash (Bobadilla et al., 2018).

6. Topological character, generalized forms, and later variants

A central development in the surface theory is that the Nash problem is topological. For normal surface singularities, the adjacency relation $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$05 depends only on the weighted dual graph of the minimal resolution, and hence, in the complex-analytic case, only on the topology of the link. This leads to reductions: if the Nash problem is true for singularities with rational homology sphere links, then it is true in general (Bobadilla, 2010).

The generalized Nash problem asks, for two divisors $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$06 and $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$07 over a normal surface $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$08, when the inclusion

$\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$09

occurs. Bobadilla, Pe Pereira, Popescu, and Pampu showed that the validity of such an adjacency depends only on the topological type of the blow-up model containing $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$10 and $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$11. They also proved a valuation-deformation correspondence,

$\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$12

with $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$13 placed on $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$14 and $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$15 placed on $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$16 for $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$17; equivalently one can take a linear deformation $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$18. This turns the generalized Nash problem into a question about adjacencies of plane curve singularities (Bobadilla et al., 2018).

Another extension is the study of holomorphic arc spaces. Kollár and Némethi introduced spaces $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$19 and $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$20 of holomorphic arcs and short arcs, together with winding number maps

$\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$21

For short arcs, this map is always injective and is bijective for $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$22-dimensional quotient singularities (Bobadilla et al., 2018).

In higher dimension, positive families beyond surfaces also exist. For normal varieties with a complexity-one torus action, Bourqui, Langlois, and Mourtada gave an explicit combinatorial description of the equivariant Nash order on $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$23-invariant divisorial valuations, solved the classical Nash problem in that setting by proving that every essential valuation is a Nash valuation, and constructed examples of Nash valuations that are neither minimal nor terminal (Bourqui et al., 2022).

The embedded Nash problem is an embedded analogue for a hypersurface $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$24, where one studies maximal irreducible families of arcs with fixed order of contact along $\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$25. For a reduced divisor, Budur, Fernández de Bobadilla, and collaborators proved the strict inclusions

$\mathcal{L}(X)=\{\gamma:\Spec \C[[t]]\to X\}$26

and solved the problem for unibranch plane-curve germs in terms of the resolution graph (Budur et al., 2022).

Taken together, these results show that the Nash problem has a sharply different behavior in dimension two and in higher dimension. For surfaces, bijectivity is complete and topological in nature. Beyond surfaces, the theory branches into terminal valuations, equivariant and embedded variants, generalized adjacency problems, and explicit counterexamples, while the search for a full higher-dimensional characterization of the image of the Nash map remains open (Bobadilla et al., 2011, Fernex et al., 2014).