Related papers: Arc spaces of cA_1 singularities

We study the space of arcs on a singularity of the form xy=f(z_1,..., z_n) and prove 2 main results. (i) The number of irreducible components equals the multiplicity of f minus 1. (ii) If n>1 and the leading homogeneous term of f is not a…

The paper surveys several results on the topology of the space of arcs of an algebraic variety and the Nash problem on the arc structure of singularities.

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.

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…

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…

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a singular germ of a surface $S$ coincides with the number of irreducible components of the exceptional divisor in the minimal…

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no…

We study loci of arcs on a smooth variety defined by order of contact with a fixed subscheme. Specifically, we establish a Nash-type correspondence showing that the irreducible components of these loci arise from (intersections of)…

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 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 $(X,O)$ 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…

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…

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…

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.

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…

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.

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of…

We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the…

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space…