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A curve selection lemma in spaces of arcs and the image of the Nash map. (English) Zbl 1118.14004

Compos. Math. 142, No. 1, 119-130 (2006); corrigendum ibid. 157, No. 3, 641-648 (2021).
The curve selection lemma for a closed point of a scheme of finite type over an algebraically closed field is trivial. But for a non-closed point of a non-noetherian scheme, it is not clear. The arc space of a variety is an example of non-noetherian scheme. In this paper, the author proves the curve selection lemma for the arc space of a variety. It says that for a generically stable point in the arc space and an irreducible subset whose closure contains the point, there is an algebra-analytic curve passing through the point and the complement of the closure of the point in the irreducible subset. By using this lemma, the author proves the equivalence between the wedge problem and the Nash problem. As an application of this equivalence one obtains the affirmative answer to the Nash problem for a sandwiched singularity, since the affirmative answer to the wedge problem for a sandwiched singularity was already proved by the author and M. Lejeune-Jalabert [Am. J. Math. 121, No. 6, 1191–1213 (1999; Zbl 0960.14015)].

MSC:

14B05 Singularities in algebraic geometry
14A15 Schemes and morphisms
32S05 Local complex singularities
32S45 Modifications; resolution of singularities (complex-analytic aspects)

Citations:

Zbl 0960.14015
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