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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.

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.

Reviewer: Peter Petrov (West Lafayette)

### MSC:

14B05 | Singularities in algebraic geometry |

32S25 | Complex surface and hypersurface singularities |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |