Nash's conjecture for normal surface singularities

Let $(X,O)$ be a normal surface singularity, and let

$$\pi:(\widetilde{X},E)\to(X,O)$$

be its minimal resolution, with exceptional divisor $E=\bigcup_{k=1}^r E_k$. For each exceptional component $E_k$, let

$$N_k=\{\gamma\in\mathcal{X}_\infty:\widetilde{\gamma}(0)\in E_k\},$$

and let $\overline{N}_k$ denote its closure in the arc space. The components $E_i$ and $E_j$ are essential divisors when they occur as irreducible components of the exceptional divisor of the minimal resolution. Nash's conjecture. For any two essential divisors $E_i$ and $E_j$, we have

$$\overline{N}_i\nsubseteq\overline{N}_j.$$

This asserts that there are no adjacencies among the families of arcs associated with distinct essential divisors, so these families give distinct irreducible components of the arc space. The conjecture is a central problem relating the geometry of resolutions of surface singularities to the structure of their arc spaces; its status is not established by the supplied source context.