## Kodai Mathematical Journal

### Families of higher dimensional germs with bijective Nash map

#### Abstract

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).

#### Article information

Source
Kodai Math. J., Volume 31, Number 2 (2008), 199-218.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1214442795

Digital Object Identifier
doi:10.2996/kmj/1214442795

Mathematical Reviews number (MathSciNet)
MR2435892

Zentralblatt MATH identifier
1210.14008

#### Citation

Plénat, Camille; Popescu-Pampu, Patrick. Families of higher dimensional germs with bijective Nash map. Kodai Math. J. 31 (2008), no. 2, 199--218. doi:10.2996/kmj/1214442795. https://projecteuclid.org/euclid.kmj/1214442795