## The local Nash problem on arc families of singularities.(English)Zbl 1116.14030

Let $$X$$ be a $$k$$-scheme. The arc space $$X_\infty$$ is characterized by the property $\text{Hom}_k(Y, X_\infty)\simeq \text{Hom}_k(Y\widehat{\times}_{\text{Spec}(k)}\text{Spec}(k[[t]]),X)$ for an arbitrary $$k$$-scheme $$Y$$. Let $$\pi_X:X_\infty\to X$$ be the canoncial projection sending an arc $$\alpha:\text{Spec}\;k[[t]]\to X$$ to $$\alpha(0)$$. A morphism $$\varphi: Z\to X$$ induces a canonical morphism $$\varphi_\infty: Z_\infty\to X_\infty$$, $$\alpha\mapsto \varphi\circ \alpha$$.
An irreducible component $$C$$ of $$\overline{\pi_X^{-1}(x)}$$ is called local Nash component of $$(X,x)$$ if $$C$$ is not contained in $$(\text{Sing }(X))_\infty$$. Let $$\varphi: Y\to X$$ be a resolution of the singularities such that $$\overline{\varphi^{-1}(x)}$$ is a union of nonsingular divisors. Let $$\overline{\varphi^{-1}(x)}=\bigcup E_j$$ be the decomposition into irreducible components. Let $$\{C_i\}$$ be the local Nash components of $$(X,x)$$. Then the morphism $$\varphi_\infty:\bigcup_j\pi_Y^{-1}(E_j)\to\bigcup_iC_i$$ is dominant and injective outside $$(\text{Sing }(X))_\infty$$. For each $$C_i$$ there is a unique $$E_{j_i}$$ such that $$\pi^{-1}_Y(E_{j_i})$$ is dominant to $$C_i$$. $$E_{j_i}$$ is an essential divisor over $$(X, x)$$. The map $l\mathcal N:\{\text{local Nash components of }(X,x)\}\to\{\text{essential divisors over }(X,x)\},$ $$l\mathcal N(C_i)=E_{j_i}$$, is called the local Nash map. This map is injective. The local Nash Problem is the question whether this map is bijective. An affirmative answer to the local Nash problem is given for toric singularities and so-called analytically pretoric singularities. This includes the class of quasi-ordinary singularities.

### MSC:

 14J17 Singularities of surfaces or higher-dimensional varieties 14M25 Toric varieties, Newton polyhedra, Okounkov bodies

### Keywords:

arc space; Nash problem; singularities
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### References:

  Bouvier, C., Diviseurs essentiels, composantes essentielle des variétés toriques singulières, Duke Math. J., 91, 609-620, (1998) · Zbl 0966.14038  Bouvier, C.; Gonzalez-Sprinberg, G., Système générateur minimal, diviseurs essentiels et G-désingularisation, Tohoku Math. J., 47, 125-149, (1995) · Zbl 0823.14006  Fulton, W., Introduction to toric varieties, 131, (1993), Princeton University Press, Princeton, NJ · Zbl 0813.14039  Gau, Y.-N., Embedded topological classification of quasi-ordinary singularities, Mem. Amer. Math. Soc., 74, 388, 109-129, (1988) · Zbl 0658.14004  Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Annals of Math., 79, 109-326, (1964) · Zbl 0122.38603  Ishii, S., The arc space of a toric variety, J. Algebra, 278, 666-683, (2004) · Zbl 1073.14066  Ishii, S., Arcs, valuations and the Nash map, J. reine angew. Math, 588, 71-92, (2005) · Zbl 1082.14007  Ishii, S.; Kollár, J., The Nash problem on arc families of singularities, Duke Math. J., 120, 3, 601-620, (2003) · Zbl 1052.14011  Lejeune-Jalabert, M.; Reguera-Lopez, A. J., Arcs and wedges on sandwiched surface singularities, Amer. J. Math., 121, 1191-1213, (1999) · Zbl 0960.14015  Lipman, J., Quasi-ordinary singularities of embedded surfaces, (1965)  Lipman, J., Singularities, Part 2 (Arcata, Calif., 1981), 40, Quasi-ordinary singularities of surfaces in $$\mbox\textbf{C}^3, 161-172, (1983),$$ Amer. Math. Soc., Providence, RI · Zbl 0521.14014  Nash, J. F., Arc structure of singularities, Duke Math. J., 81, 31-38, (1995) · Zbl 0880.14010  Oh, K., Topological types of quasi-ordinary singularities, Proc. AMS, 117, 53-59, (1993) · Zbl 0791.32018  Pérez, P. D. Gonález, Toric embedded resolutions of quasi-ordinary hypersurface singularities, Ann. Inst. Fourier (Grenoble), 53, 1819-1881, (2003) · Zbl 1052.32024  Plenat, C.; Popescu-Pampu, P., A class of non-rational surface singularities for which the Nash map is bijective · Zbl 1119.14007  Reguera, A. J., Image of Nash map in terms of wedges, C. R. Acad. Sci. Ser. I, 338, 385-390, (2004) · Zbl 1044.14032  Reguera-Lopez, A. J., Families of arcs on rational surface singularities, Manuscr. Math., 88, 321-333, (1995) · Zbl 0867.14012  Vojta, P., Jets via Hasse-Schmidt derivations · Zbl 1194.13027
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