Families of higher dimensional germs with bijective Nash map

Abstract

Let ðX ; 0Þ be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over ðX ; 0Þ is a divisorial valuation of the field of meromorphic functions on ðX ; 0Þ, 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 p of ðX ; 0Þ such that its reduced exceptional divisor carries su‰ciently many p-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).

The Nash problem for surfaces

“…In the same paper they showed the bijectivity of the Nash mapping for toric singularities of arbitrary dimension. Other advances in the higher dimensional case include [23], [6], [14]. Very recently there have appeared 3-dimensional counterexamples as well.…”

Terminal valuations and the Nash problem

“…However, X ∞ is not a Noetherian scheme, and there are examples where the curve selection lemma fails in the non-Noetherian setting (e.g., see [24,Example 4]). It is therefore a delicate issue to establish the existence of a wedge with the above properties.…”

The Nash Problem from Geometric and Topological Perspective

“…In the same paper they showed the bijectivity of the Nash mapping for toric singularities of arbitrary dimension. Other advances in the higher dimensional case include [25], [6], [16]. In 2013 T. de Fernex [1] found the first counter-examples to the Nash question; further counterexamples, and a deeper understanding of how they appear was provided by J. Johnson and J. Kollár in [7].…”