Families of higher dimensional germs with bijective Nash map

Kodai Mathematical Journal

Camille Plénat²  Patrick Popescu-Pampu¹ 

[1] Univ. Paris 7 Denis Diderot Inst. de Maths.-UMR CNRS 7586 équipe "Géométrie et dynamique";Université de Provence LATP UMR 6632 Centre de Mathématiques et Informatique

关键词: Space of arcs;    essential components;    Nash map;    Nash problem;   

DOI  :  10.2996/kmj/1214442795

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(25)Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution π of (X,0) such that its reduced exceptional divisor carries sufficiently many π-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).

【 授权许可】

Unknown