## 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).
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

Zbl 0607.00005