Topology of the quasi-ordinary surface singularities

Abstract

IN this paper we are interested in one type of (local) complex surface singularities (nonisolated in general, (2.4.3) ex. 2) the investigation of which is due essentially to Lipman [20]. Let F be the surface in an open set in C3 defined by the vanishing of a holomorphic function in three variables. Assume that the origin 0 is a singular (nonsmooth) point of F. If there is a projection of F to a complex plane so that the discriminant (branch) curve has a smooth point at the image of 0, then F is equisingular at 0 (along the preimage of the discriminant curve) [27]. The surface singularities that we are interested in are those which are next in order of complexity, namely those whose images in some (local) projections are ordinary double points of the discriminant curves. Such singularities are said to be quasi-ordinary or Jungian (2.1). See [21] for an introduction to the quasi-ordinary singularities and a summary of the fundamental work [20]. Quasi-ordinary singularities are amenable to detailed analysis since they can be “parametrized” by fractional power series in two variables [19, p. 2071, [Z], which are called the parametrizations of the singularities (cf. the irreducible plane curve singularities can always be parametrized by fractional power (Puiseux) series in one variable). From a suitably normalized parametrization of a quasi-ordinary singularity one can single out (in the same way one obtains the characteristic pairs of plane curves) some exponents. called the distinguished pairs, which are shown in [20] to depend only on the analytic type ( = isomorphism type of the analytic local ring) of the singularity. Moreover, like the characteristic pairs for plane curves, the distinguished pairs determine quite a lot of geometry of the singularity [21, p. 1631. The main result of this paper is that the distinguished pairs of (F, 0) depend only on the topological type of (C3, F) at 0. In other words, the local topology of a quasi-ordinary singularity determines its distinguished pairs. Since the distinguished pairs determine the local topology (2.2.3), it follows that topo!ogically the quasi-ordinary singulariries are completely classified by rheir disringuished pairs (2.2.4). As a corollary, the question of Zariski about the topological invariance of multiplicity is answered affirmatively for quasi-ordinary singularities (2.2.5). Another consequence of the main result is that the local topology determines the Zariski tangent cone (up to isomorphism) (2.2.5)-ii, this answers a question of L(LTeissier [ 18, p. 107]in the quasi-ordinary case. The following question may be of some interest to topologists: Question (Lipman [ZO, p. 1683). Is there an explicit topological interpretation of the distinguished pairs, as one has via compound torus knots for the characteristic pairs of plane curves? The main result: “local topology determines the distinguished pairs (2.2.1)” will be proved