Topological types and multiplicities of isolated quasi-homogeneous surface singularities

Abstract

ABSTRACT. TWO germs of 2-dimensional isolated quasi-homogeneous hypersurface singularities have the same topological type if and only if they have the same characteristic polynomial and the same fundamental group for their links. In particular, multiplicity is an invariant of topological type, an affirmative answer to Zariski's question in this case.Let (V, 0) and (W, 0) be germs of isolated hypersurface singularities in C n+1 . We say that (V, 0) and (W,0) have the same topological type if there is a germ of a homeomorphism from (C n+1 , V,0) to (C n+1 , W, 0). In his retiring Presidential address to the American Mathematical Society in 1971, Zariski asked whether (V, 0) and {W,0) have the same multiplicity if they have the same topological type. He expected that topologists would be able to answer his question in relatively short order. However the question appears to be much harder than what Zariski thought. Even special cases of Zariski's problem have proved to be extremely difficult. Only recently Greuel [4] and O'Shea [14] proved independently that topological type constant families of isolated quasi-homogeneous singularities are equimultiple. For quasihomogeneous surface singularities, Laufer [5] explained the constant multiplicity for a topological type constant family of singularities from a different viewpoint. However it is not known whether two quasi-homogeneous singularities having the same topological type can be put into a topological type constant family. Let (V, 0) be a dimension two isolated hypersurface singularity. Lê and Teissier [8] observed that A'Campo's work [1] can often be used to give positive results towards Zariski's question. Let C(V,0) be the reduced tangent cone. Let PC(V, 0) denote the hypersurface in CP 2 over which C(V, 0) is a cone. Then, the work of A'Campo shows that the multiplicity of (V, 0) is determined by the topological type of (V, 0) in case the topological Euler number x(PC(^0)) is nonzero. The same arguments also show that, for isolated hypersurface two-dimensional singularities, the embedded topology and the multiplicity determine x(PC(^0)))-However, so far, by using A'Campo's result, one can only prove that a surface in C 3 having at 0 a singularity of multiplicity 2 cannot have the same topological type at 0 as another surface of multiplicity different from 2.

The Łojasiewicz exponent of a set of weighted homogeneous ideals

“…That is, L 0 (f ) depends only on the weights w i and the degree d in this case. Therefore it is concluded that L 0 (f ) is a topological invariant of f , by virtue of the results of Saeki [15] and Yau [20]. In view of the above equality it is reasonable to conjecture that the analogous result holds in general, that is, if f : (C n , 0) → (C, 0) is a weighted homogeneous polynomial, or even a semi-weighted homogeneous function (see Definition 4.1), with respect to (w 1 , .…”

Motivic-type invariants of blow-analytic equivalence

“…This difficulty is overcomed thanks to the Lojasiewicz's theory of relatively semi-algebraic, semi-analytic sets [27]. Thom [31], [33], [36], [37] [20], [21] [8], [7], [9]. Instead [16], [3].…”

Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

“…We remark that when d 2w i , for all i = 1, 2, 3, then L 0 (∇f ) = d−w 0 w 0 . As a consequence of (3) we have that if f : C 3 → C is a weighted homogeneous function with respect to (w 1 , w 2 , w 3 ), then L 0 (∇f ) is a topological invariant of f , by the results of Saeki [27] and Yau [32].…”