For an algebraic variety \(X\), the Nash problem is asking if the set of Nash components over \(X_{\mathrm{sing}}\) in its arc space \(X_\infty\), and the set of essential divisors over \(X\) are in bijection by the natural Nash map between them, which is always injective [S. Ishii and J. Kollár, Duke Math. J. 120, No. 3, 601–620 (2003; Zbl 1052.14011)]. The answer is positive for \(A_n\) [J. F. Nash, jun., Duke Math. J. 81, No.1, 31–38 (1995; Zbl 0880.14010)], for \(D_n\) [C. Plénat, C. R., Math., Acad. Sci. Paris 340, No. 10, 747–750 (2005; Zbl 1072.14004)], for sandwiched surface singularities [M. Lejeune-Jalabert and A. J. Reguera-López, Am. J. Math. 121, No. 6, 1191–1213 (1999; Zbl 0960.14015)], for toric varieties [Zbl 1052.14011], and for some other classes. From [J. F. de Bobadilla and M. Pe Pereira, “Nash problem for surfaces”, arXiv:1102.2212], it is positive for normal surfaces, in particular for \(\mathbb E_6, \mathbb E_7, \mathbb E_8\) singularities (see also C. Plénat and M. Spivakovsky, “The Nash problem of arcs and the rational double point \(\mathbb E_6\)”, arXiv:1011.2426]). There are counter examples in dimension 4 [Zbl 1052.14011], and recently, in dimension 3 [T. de Fernex, “Three-dimensional counter-examples to the Nash problem”, arXiv:1205.0603].
It is known that Nash problem for normal surface singularities could be reduced to the case of quasi-rational singularities, i.e. such that the divisors appearing in the minimal resolution are all rational. In the article under review is developed an approach to solve it for a family of quasi-rational surfaces \(S(p, h_q ) : z^p + h_q (x, y) = 0\), where hq is a degree q reduced homogeneous polynomial, and \(p, q \geq 2\) are coprime. Then using the same method is obtained positive answer for \(\mathbb E_6, \mathbb E_7\) and \(D_n\) surface singularities. It works in higher dimensions as well, as demonstrated in the author’s thesis, providing new examples of dimension 3 hypersurface singularities having bijective Nash map (http://tel.archives-ouvertes.fr/tel-00630100).
The main theorem is claiming that for any \(p, q\) as above, the Nash problem for \(S(p, h_q )\) has a positive answer. To prove it, an embedded resolution of \(S(p, h_q )\) is constructed first. The cases to be checked could be reduced now to \(p > q \geq 3\), and coprime. The idea is to apply the theorem of Lejeune- Jalabert and Reguera [A. J. Reguera, Compos. Math. 142, No. 1, 119–130 (2006; Zbl 1118.14004)], that for a given resolution\( f :\tilde X \to X\) an essential component \(E \subset \tilde X\) belongs to the image of the Nash map iff any wedge over \(X\) whose special arc is the generic point of the family \(N_E \subset X_{\infty}\), corresponding to E, and whose generic arc is over \(X_{\mathrm{sing}}\), could be lifted to \(\tilde X\). The key proposition needed is that when \(\tilde X\) is the minimal resolution for \(S(p, h_q )\), for a given wedge this holds if the three associated with the wedge power series in \(K[[s, t]]\) are invertible. Finally it is shown their invertibility for any wedge as above.
With the same method the Nash problem is then solved positively for \(\mathbb E_6\) and \(\mathbb E_7\) surface singularities. The approach here is different from the topological one accepted by de Bobadilla and Pereira, whose paper appeared after this article was submitted. At the end with this method is obtained also a new and simpler proof of the problem for the case of \(D_n\) surface singularities, \(n \geq 4\).
In this article the ground field is always assumed to be of characteristic \(0\).