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Date(s) - 6 décembre 2012

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{{{}}} Let – : ( ~X ;E) ! (X;O) be a resolution of singularities of a singularity (X;O). Take the decomposition of the exceptional divisor E = S i Ei. Given any arc : (C; 0) ! (X; SingX) one can consider the lifting ~ : (C; 0) ! ( ~X ;E). Nash considered the set of arcs whose lifting ~ meets a -x divisor Ei, that is ~ (0) 2 Ei, and proved that its closure is an irreducible set of the space of arcs. Nash’s question is whether for the essential divisors Ei these subsets of arcs are in fact irreducible components of the space of arcs or not. He conjectured that the answer was yes for the case of surfaces and suggested the study in higher dimensions. Recently we solved the conjecture for the surface case in a joint work with J. Fern-andez de Bobadilla. I will give an introduction to the problem and details of the proof for the normal surface case.