Monodromy conjecture for some surface singularities

Abstract

ABSTRACT. -In this work we give a formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three.

Monodromy Jordan blocks, b-functions and poles of zeta functions for germs of plane curves

“…Conjecture 3 for the local topological zeta function Z top,0 ( f , s) of a SIS singularity has been proved by Artal Bartolo, Cassou-Noguès, Luengo and the first author in [3].…”

On monodromy for a class of surfaces

“…Loeser a demontré la conjecture de monodromie pour les courbes et pour les polynômes qui sont nondégénérés pour leur polyèdre de Newton et qui satisfont des conditions numériques [6,7]. Artal-Bartolo, Cassou-Noguès, Luengo, Melle-Hernández [2], Rodrigues [8] et Veys [9] ont également traité de cette conjecture mystérieuse.…”

“…Loeser proved the conjecture for curves and for polynomials that are nondegenerate with respect to their Newton polyhedron and that satisfy some numerical conditions [6,7]. Also Artal-Bartolo, Cassou-Noguès, Luengo, Melle-Hernández [2], Rodrigues [8] and Veys [9] provided results about this mysterious conjecture. We explain in this Note by geometric arguments a phenomenon concerning calculation of monodromy and the monodromy conjecture for surfaces that are generic with respect to a 3-dimensional toric idealistic cluster.…”

Links and analytic invariants of superisolated singularities

“…[18], page 135.) If d = 3, then C has a unique singularity of type [2]. If d = 4, then there are four possibilities; the corresponding multiplicity sequences of the singular points {p i } i of C are [3]; [2 3 [2] and [2], [2], [2].…”