arXiv math/06096292006-09-22math.AGmath.AC

曲面奇性圆弧上的Nash问题

The Nash problem on arcs for surface singularities

Marcel Morales

备注：: 17 pages, 5 figures

摘要：

设$(X,O)$是法向曲面奇点的芽，$π: \tilde X\longrightarrow X$是奇点的最小分解，$A=(a_{i,j})$是$\tilde X$的例外集的$n\times n$对称交矩阵.在一个旧的预印本纳什证明，一套弧的表面奇异性是一个计划${\cal H}$，并定义了一个地图${\cal N}$从一套不可约的组成部分${\cal H}$的一套例外的组成部分的最小分辨率的奇异性$(X,O)$。他证明，这张地图是内射，并要求如果它是满射。本文研究了正则分解${\cal H}=\cup_{i=1}^n \bar{\cal N}_i$：0对于任意两个不同例外分支的$(E_i,E_j)$，定义了数值Nash条件$(NN_{(i,j)})$ .我们知道$(NN_{(i,j)})$蕴涵$\bar{\cal N}_{i}\not\subset \bar{\cal N}_{j} $。在本文中，我们证明了$(NN_{(i,j)})$总是正确的至少有一半的夫妇$(i,j)$。条件$(NN_{(i,j)})$对于所有的$(i,j)$与$i\not=j$对都成立，它描述了一类负定矩阵，我们称之为纳什矩阵。如果$A$是纳什矩阵，则纳什映射${\cal N}$是双射的。特别是我们的结果只依赖于$A$，而不是例外集的拓扑类型。o我们恢复和改善相当大的几乎所有已知的结果在这个主题和我们的证明是新的和基本的。我们给出了无穷多个其他类型的奇点，其中纳什猜想是真的。证明是基于我的老作品\cite{M}和Plenat \cite{P}。

英文摘要：

Let $(X,O)$ be a germ of a normal surface singularity, $π: \tilde X\longrightarrow X$ 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 ${\cal H}$, and defines a map ${\cal N}$ from the set of irreducible components of ${\cal H}$ 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 ${\cal H}=\cup_{i=1}^n \bar{\cal N}_i$: o 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{\cal N}_{i}\not\subset \bar{\cal N}_{j} $. In this paper we prove that $(NN_{(i,j)})$ is always true for at least the half of couples $(i,j)$. o 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 ${\cal N}$ is bijective. In particular our results depends only on $A$ and not on the topological type of the exceptional set. o We recover and improve considerably almost all results known on this topic and our proofs are new and elementary. o We give infinitely many other classes of singularities where Nash Conjecture is true. The proofs are based on my old work \cite{M} and in Plenat \cite{P}.

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