The Nash problem of arcs and the rational double points D_n

The Nash problem for surfaces

“…On the other hand, bijectivity of the Nash mapping has been shown for many classes of surfaces (see [6], [10], [8], [9], [12], [13], [17], [19], [20], [21], [22], [24], [25], [26]). The techniques leading to the proof of each of these cases are different in nature, and the proofs are often complicated.…”

“…The techniques leading to the proof of each of these cases are different in nature, and the proofs are often complicated. It is worth noting that even for the case of the rational double points not solved by Nash, a complete proof had to be awaited until 2010; see [19], where the problem is solved for any quotient surface singularity, and also [21] and [24] for the cases of D n and E 6 . In [4] it is shown that the Nash problem for surfaces only depends on the topological type of the singularity.…”

Nash problem for quotient surface singularities

“…Using the shape of the resolution graphs of the quotient surface singularities given in [2] and a comparison theorem of [5], we can reduce the proof of Theorem 1.1 to the singularities D n and E 8 . The singularities D n were already settled in [27] and we give another proof of it in [24] using our method. We include here the complete proof for the E 8 singularity.…”

“…For a more extended introduction to the problem, one can see [23] or [15].In dimension 4, Ishii and Kollár [15] found an example of singularity with non-bijective Nash mapping. On the other hand, the bijectivity of the Nash mapping is still open for surfaces and has been proved in many classes of singularities: A k singularities [23]; normal minimal surface singularities [31] (and also [7,26]); sandwiched singularities [19,32]; the dihedral singularities D n (see [27]); a family of non-rational surface singularities in [28]; toric singularities [15]; quasi-ordinary singularities [8,14]; stable toric varieties [25] and some other families of higherdimensional singularities [29], among others.In addition, there are some papers proving reductions of the problem. For instance, in [18] it is proved that divisors that are not uniruled are in the image of the Nash map and that the so-called lifting wedge property for the case of quasi-rational surfaces would imply that the surjectivity of the Nash map for normal surface singularities.…”

Familles d'espaces de m -jets et d'espaces d'arcs

“…Étant donné explicitement une variété V et une désingularisation π, on remarque que déterminer si une composante irréductible de la fibre exceptionnelle de π est ou non un diviseur essentiel reste un problème difficile. L'approche de Nash est donc très De nombreux mathématiciens, qu'on ne peut pas tous citer ici, ont apporté de nouvelles contributions originales à l'étude du problème de Nash, surtout dans le cas de variétés de dimension 2 et 3 (voir par exemple [LJ80], [Reg95], [GSLJ97], [LJRL99], [IK03], [Ish05], [Ish06], [PPP06], [GP07], [Mor08], [Plé08], [PPP08], [Pet09], [LA11], [FdB12], [PS12], [dFD14]). …”