Related papers: The Nash problem of arcs and the rational double p…

The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ (S,O) of a singular surface as there are essential divisors over (S,O). It is known that this problem can be reduced to the study…

The embedded Nash problem for a hypersurface in a smooth algebraic variety, is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal…

We prove that the Nash problem holds for two-dimensional rational double points in all characteristics. The proof is based on a direct computation of the families of arcs through these singularities.

This paper deals with the Nash problem, which claims that there are as many families of arcs on a singular germ of surface $U$ as there are essential components of the exceptional divisor in the desingularisation of this singularity. Let…

In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one.…

Nash proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that occurs on every resolution. He asked if the converse also holds: does every such exceptional divisor…

Let (S,0) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E_i. The Nash map associates to each irreducible component C_k of the space of…

Let $(X,O)$ be a germ of a normal surface singularity, $\pi : \tilde X\longrightarrow X$ 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…

This paper seeks to prove the bijectivity of the "Nash mapping" from the set of irreducible components of the scheme parametrizing analytic arcs on an algebraic surface $X$ whose origin is a singular point, into the set of irreducible…

We address Nash problem for surface singularities using wedges. We give a refinement of the characterisation of A. Reguera of the image of the Nash map in terms of wedges. Our improvement consists in a characterisation of the bijectivity of…

This paper shows the affirmative answer to the local Nash problem for a toric singularity and analytically pretoric singularity. As a corollary we obtain the affirmative answer to the local Nash problem for a quasi-ordinary singularity.

Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of…

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…

We prove that, if X is a variety over an uncountable algebraically closed field k of characteristic zero, then any irreducible exceptional divisor E on a resolution of singularities of X which is not uniruled, belongs to the image of the…

The Nash problem asks about the existence of a correspondence between families of arcs through singularities of complex varieties and certain types of divisorial valuations. It has been positively settled in dimension 2 by Fern\'andez de…

In this paper we give a positive answer to a question of Nash concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities, extending to this case previous results and techniques of Shihoko Ishii.

This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see…

We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the authors influenced it. Later we summarize the main ideas in the higher dimensional statement and…

Let p be a singular point of a complex hypersurface whose tangent cone is a quadric of rank at least 3. We show that the space of arcs through p is irreducible. Using a method of de Fernex, this shows that the Nash problem has a negative…

We consider the question whether one can construct an embedded resolution of singularities of a singular variety $X\subset \textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular…