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**On the resolution of the hypersurface singularities.**
*(English)*
Zbl 0622.14012

Complex analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 405-436 (1987).

[For the entire collection see Zbl 0607.00005.]

Let \(f(z_ 0,...,z_ n)\) be a germ of an analytic function with an isolated critical point at the origin; f is assumed to have a non- degenerate Newton boundary \(\Gamma\) (f). Let V be the germ of the hypersurface \(f^{-1}(0)\) at the origin. There is always a canonical resolution \(\pi: \tilde V\to V\) of V which is associated with a given simplicial subdivision \(\Sigma^*\) of the dual Newton diagram \(\Gamma^*(f)\). The main purpose of this paper (see in particular § 5) is to study the topology of the exceptional divisors E(P) through a canonical simplicial subdivision \(\Sigma^*\), which is constructed in § 3. If P is a strictly positive vertex of \(\Sigma^*\), then E(P) is always a compact divisor such that \(\pi (E(P))=\{0\}\). The topology of exceptional divisors E(P) of the two dimensional and the three dimensional singuarities is then studied in detail in §§ 6 and 8 respectively. In section \(7\) the author shows that the fundamental group of E(P), where P is a strictly positive vertex of a fixed subdivision \(\Sigma^*\), is a finite cyclic group (with an order independent of the choice of \(\Sigma^*)\) if \(n>2\) and if the face \(\Delta\) (P) of \(\Gamma\) (f) is an n-simplex. In section \(9\) the canonical divisors of \(\tilde V\) and E(P) are considered. In particular, applying (9.1) and (9.2) one can calculate the signature of the Milnor fibre of f in the case \(n=2\) from the Newton boundary \(\Gamma\) (f).

Let \(f(z_ 0,...,z_ n)\) be a germ of an analytic function with an isolated critical point at the origin; f is assumed to have a non- degenerate Newton boundary \(\Gamma\) (f). Let V be the germ of the hypersurface \(f^{-1}(0)\) at the origin. There is always a canonical resolution \(\pi: \tilde V\to V\) of V which is associated with a given simplicial subdivision \(\Sigma^*\) of the dual Newton diagram \(\Gamma^*(f)\). The main purpose of this paper (see in particular § 5) is to study the topology of the exceptional divisors E(P) through a canonical simplicial subdivision \(\Sigma^*\), which is constructed in § 3. If P is a strictly positive vertex of \(\Sigma^*\), then E(P) is always a compact divisor such that \(\pi (E(P))=\{0\}\). The topology of exceptional divisors E(P) of the two dimensional and the three dimensional singuarities is then studied in detail in §§ 6 and 8 respectively. In section \(7\) the author shows that the fundamental group of E(P), where P is a strictly positive vertex of a fixed subdivision \(\Sigma^*\), is a finite cyclic group (with an order independent of the choice of \(\Sigma^*)\) if \(n>2\) and if the face \(\Delta\) (P) of \(\Gamma\) (f) is an n-simplex. In section \(9\) the canonical divisors of \(\tilde V\) and E(P) are considered. In particular, applying (9.1) and (9.2) one can calculate the signature of the Milnor fibre of f in the case \(n=2\) from the Newton boundary \(\Gamma\) (f).

Reviewer: M.Herrmann

### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

14J17 | Singularities of surfaces or higher-dimensional varieties |

32S05 | Local complex singularities |

14F45 | Topological properties in algebraic geometry |