Forme de Jordan de la monodromie des singularités superisolées de surfaces

Partial resolutions and the zeta-function of a singularity

“…. (f m is homogeneous of degree m) with f d and f d+k satisfying some special conditions which has been extensively studied: E. Artal-Bartolo [3], I. Luengo [4], A. Melle-Hernández [5], D. Siersma [6], J. Stevens [7]. In particular we show how the formula of D. Siersma, [6] and J. Stevens, [7], for the zeta-function (or rather for the characteristic polynomial in their papers) of some isolated singularities of the form f = f d + f d+k can be obtained in this way.…”

Monodromy conjecture for some surface singularities

“…see [2] -and the monodromy conjecture for curvessee [21] -we prove the following for a SIS singularity of multiplicity d:…”

“…We prove, case by case, that exp(2iπ(−3/d)) is a root of the Alexander polynomial of the curve at its only singular point with the required multiplicity. Finally, using the computations in [2], we have that exp(2iπ(−3/d)) is a root of the Alexander polynomial of the corresponding SIS singularity.…”

“…An embedded resolution is also known for superisolated surface singularities, SIS for short -see [2]. Singularities of this type, named by I. Luengo in [25], were used to prove that the µ-constant stratum of an isolated hypersurface singularity is not smooth -see also [28].…”

Links and analytic invariants of superisolated singularities

“…They were introduced in [16] by the first author in order to show that the µ-constant stratum in the semiuniversal deformation space of an isolated hypersurface singularity, in general, is not smooth. Later Artal-Bartolo in [1] used them to provide a counterexample for S. S.-T. Yau's conjecture (showing that, in general, the link of an isolated hypersurface surface singularity and its characteristic polynomial not determine the embedded topological type of the singular germ). On the other hand, A. Durfee's conjecture and the monodromy conjecture of J. Denef and F. Loeser has been proved for them, see [17] and [2].…”