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A class of non-rational surface singularities with bijective Nash map. (English) Zbl 1119.14007

Let \((S, 0)\) be a germ of complex analytic normal surface singularity with \(\{E_i\}_{i \in I}\) the irreducible components of the exceptional divisor \(E\) in its minimal resolution \(\pi_m: ({\tilde{S}}_m, E) \rightarrow (S, 0)\). The Nash map associates to each irreducible component \(C_j\) of the pre-image of \(0\) in the space of arcs a unique divisor \(E_{i(j)}\), and it is an injective map [J. F. Nash, Jr., Duke Math. J. 81, No. 1, 31–38 (1996; Zbl 0880.14010)]. The Nash problem asks (in arbitrary dimension) for which classes of singularities it is bijective. There is a counter example in dimension 4 [S. Ishii and J. Kollár, Duke Math. J. 120, No. 3, 601–620 (2003; Zbl 1052.14011)], which generalizes easily in higher dimensions, but in dimensions 2 and 3 the problem still has an answer only in some special cases.
For surfaces these include \(A_n\) singularities [J. F. Nash, Jr., Duke Math. J. 81, No. 1, 31–38 (1996; Zbl 0880.14010)], \(D_n\) singularities [C. Plénat, C. R. Math. Acad. Sci. Paris 340, No. 10, 747–750 (2005; Zbl 1072.14004)], minimal [A.-J. Reguera, Manuscr. Math. 88, No. 3, 321–333 (1995; Zbl 0867.14012)] and sandwiched singularities [M. Lejeune-Jalabert and A. J. Reguera-López, Am. J. Math. 121, No. 6, 1191–1213 (1999; Zbl 0960.14015); A. J. Reguera, C. R. Math. Acad. Sci. Paris 338, No. 5, 385–390 (2004; Zbl 1044.14032)]. In a recent preprint [M. Morales, arXiv:math.AG/0609629], infinitely many classes of surface singularities for which the Nash problem has a positive answer are constructed, and some known results are improved. Until now there is no example giving a negative answer to the problem in dimensions 2 and 3.
The main result in this paper is the following one. If in the vector space with basis \(\{ E_i \}_{i \in I}\) each open half-space \(\{ \sum_i a_i E_i| ~a_i<a_j \}\) contains a divisor \(D \neq 0\), supported on the exceptional locus and such that \(D.E_i<0 ~\forall i \in I\), then the Nash problem has a positive answer for \((X, 0)\). This is a condition only on the intersection matrix of \(\pi_m\), but does not depend on the genera or smoothness of \(E_i\). The construction is based on two results. The first one is a sufficient criterion [C. Plénat, Ann. Inst. Fourier 55, No. 3, 805–823 (2005; Zbl 1080.14021)] (in arbitrary dimension), proposed for rational surface singularities in [A.-J. Reguera, Manuscr. Math. 88, No. 3, 321–333 (1995; Zbl 0867.14012)], which helps to determine if some \(E_i\) is in the image of the Nash map. This criterion plays a central role in almost all results obtained in dimension 2. The second one is a numerical criterion, which proof is based on a result of H. B. Laufer [Am. J. Math. 94, 597–608 (1972; Zbl 0251.32002)], and gives a sufficient condition for an effective divisor with support on the exceptional locus of \(\pi_m\) to be the exceptional part of a regular function on \((S, 0)\). It is related to another result of H. B. Laufer [in: Singularities. Proc. Sympos. Pure Math. 40, 1–29 (1983; Zbl 0568.14008)], but neither one follows from the other. As a corollary of the main result one has that if \(E.E_i <0 ~\forall i\), then the Nash map is bijective. Another corollary gives infinitely many families of pairwise topologically distinct non-rational surface singularities, for which the Nash problem has a positive answer.
Reviewer’s remark: This paper is an important contribution to the list of classes of varieties, for which the answer of the Nash problem is known. I think it is well organized and easy to understand. Similar results are obtained in the preprint of Morales cited above, but the differences are not clearly explained.

MSC:

14B05 Singularities in algebraic geometry
32S25 Complex surface and hypersurface singularities
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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References:

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