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