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

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.

The proofs are based on my old work [8] and in Plenat [10].

Citation

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

Information

Published: 2008
First available in Project Euclid: 17 September 2008

zbMATH: 1178.14004
MathSciNet: MR2451219

Subjects:
Primary: 14B05 , 14E15 , 14J17 , 32Sxx

Rights: Copyright © 2008 Editorial Board, Nagoya Mathematical Journal

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