Nagoya Mathematical Journal

Some numerical criteria for the Nash problem on arcs for surfaces

Marcel Morales

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Let $(X, O)$ be a germ of a normal surface singularity, $\pi : \tilde X \to X$ be the minimal resolution of singularities and let $A = (a_{i, j})$ be the $n \times n$ symmetrical intersection matrix of the exceptional set of $\tilde X$. In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme $\mathcal{H}$, and defines a map $\mathcal{N}$ from the set of irreducible components of $\mathcal{H}$ to the set of exceptional components of the minimal resolution of singularities of $(X, O)$. He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition $\mathcal{H} = \bigcup_{i=1}^{n} \bar{\mathcal{N}_{i}}$:

  • For any couple $(E_{i}, E_{j})$ of distinct exceptional components, we define Numerical Nash condition $(NN_{(i, j)})$. We have that $(NN_{(i, j)})$ implies $\bar{\mathcal{N}_{i}} \not\subset \bar{\mathcal{N}_{j}}$. In this paper we prove that $(NN_{(i, j)})$ is always true for at least the half of couples $(i, j)$.
  • The condition $(NN_{(i, j)})$ is true for all couples $(i, j)$ with $i \not= j$, characterizes a certain class of negative definite matrices, that we call Nash matrices. If $A$ is a Nash matrix then the Nash map $\mathcal{N}$ is bijective. In particular our results depend only on $A$ and not on the topological type of the exceptional set.
  • We recover and improve considerably almost all results known on this topic and our proofs are new and elementary.
  • We give infinitely many other classes of singularities where Nash Conjecture is true.
The proofs are based on my old work [8] and in Plenat [10].

Article information

Nagoya Math. J., Volume 191 (2008), 1-19.

First available in Project Euclid: 17 September 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14J17: Singularities [See also 14B05, 14E15] 32SXX


Morales, Marcel. Some numerical criteria for the Nash problem on arcs for surfaces. Nagoya Math. J. 191 (2008), 1--19.

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