[Paper Review] The Nash problem of arcs and the rational double point $\mathbf{E_6}$

Camille Plénat, Mark Spivakovsky|arXiv (Cornell University)|Nov 10, 2010

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper provides a complete solution to the Nash problem for the rational double point $\mathbf{E_6}$, proving that the number of families of arcs on the singularity equals the number of irreducible components of the exceptional divisor in its minimal resolution. The authors establish a general strategy using a valuative criterion and detailed algebraic analysis of jet spaces and wedge equations, successfully resolving all required non-inclusions for $\mathbf{E_6}$.

ABSTRACT

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a singular germ of a surface $S$ coincides with the number of irreducible components of the exceptional divisor in the minimal resolution of this singularity. We propose a program for an affirmative solution of the Nash problem in the case of normal 2-dimensional hypersurface singularities. We illustrate this program by giving an affirmative solution of the Nash problem for the rational double point $\mathbf{E_6}$. We also prove some results on the algebraic structure of the space of $k$-jets of an arbitrary hypersurface singularity and apply them to the specific case of $\mathbf{E_6}$.

Motivation & Objective

• To solve the Nash problem affirmatively for the rational double point $\mathbf{E_6}$, which asks whether the number of arc families equals the number of exceptional divisor components.
• To develop a general program applicable to normal 2-dimensional hypersurface singularities, using algebraic and valuative techniques.
• To prove that the irreducible components of the space of arcs are precisely the closures $\overline{N_i}$, corresponding to each exceptional divisor $E_i$.
• To extend the solution to singularities with dual graphs derived from $\mathbf{E_6}$ by increasing self-intersection weights.

Proposed method

• Apply a valuative criterion: if $\mathrm{ord}_{E_i}(f) < \mathrm{ord}_{E_j}(f)$ for some $f \in \mathcal{O}_{S,0}$, then $\overline{N_i} \not\subset \overline{N_j}$.
• Use the Lipman cone to identify integer-valued cycles with distinct multiplicities, ensuring at least one non-inclusion per pair of divisors.
• Analyze the algebraic structure of $k$-jets of the singularity to study arc components via jet schemes and wedge equations.
• Construct and solve a system of wedge equations derived from the defining equations of the $\mathbf{E_6}$ singularity in a parametrized form.
• Use computational algebra (e.g., XMaple) to analyze consistency of the wedge systems and derive contradictions when assumptions violate the non-inclusion conditions.
• Systematically eliminate all possible cases of $\overline{N_i} \subset \overline{N_j}$ by contradiction, showing no consistent solution exists under such assumptions.

Experimental results

Research questions

• RQ1 Does the Nash map from arc families to exceptional divisors of the minimal resolution of $\mathbf{E_6}$ singularities remain surjective?
• RQ2 Can the non-inclusion $\overline{N_i} \not\subset \overline{N_j}$ be established for all pairs $i \neq j$ in the $\mathbf{E_6}$ case?
• RQ3 Is the general strategy based on the valuative criterion and jet space analysis sufficient to resolve the Nash problem for $\mathbf{E_6}$?
• RQ4 Can the solution for $\mathbf{E_6}$ be extended to singularities with modified dual graphs, such as increased self-intersection numbers?

Key findings

• The Nash problem has an affirmative answer for the rational double point $\mathbf{E_6}$, meaning the number of arc families equals the number of exceptional divisor components.
• All required non-inclusions $\overline{N_i} \not\subset \overline{N_j}$ for $i \neq j$ are established, with the valuative criterion resolving half and algebraic wedge analysis resolving the rest.
• The system of wedge equations derived from the $\mathbf{E_6}$ singularity equations leads to contradictions when assuming $\overline{N_i} \subset \overline{N_j}$, proving such inclusions cannot hold.
• The solution for $\mathbf{E_6}$ implies the Nash problem also holds for all singularities with dual graphs obtained by increasing the self-intersection numbers of $\mathbf{E_6}$.
• The algebraic structure of the $k$-jet spaces of the $\mathbf{E_6}$ singularity is analyzed in detail, enabling the construction of consistent arc parametrizations.