Jordan form distinction from Alexander polynomial difference for g_j = z1^2 f_j + h_j singularities

Prove that for the surface singularity germs g_0(z_1,z_2,z_3) = z_1^2 f_0(z_1,z_2,z_3) + h_0(z_2,z_3) and g_1(z_1,z_2,z_3) = z_1^2 f_1(z_1,z_2,z_3) + h_1(z_2,z_3) in C^3, where f_0 and f_1 are reduced, irreducible, homogeneous polynomials of the same degree d whose projective plane curves C_0 = V_{P^2}(f_0) and C_1 = V_{P^2}(f_1) form a Zariski pair, and h_0 and h_1 are convenient homogeneous polynomials in z_2 and z_3 of degree d+3 chosen so that z_1^2 f_j(0,z_2,z_3)+h_j(z_2,z_3) is Newton non-degenerate and Sing(V_{P^2}(f_j)) ∩ V_{P^2}(h_j) = ∅, if the Alexander polynomials of C_0 and C_1 are different, then the Jordan normal form of the monodromy of g_0 at the origin is different from that of g_1, and therefore the surface germs (V(g_0),0) and (V(g_1),0) have distinct embedded topologies in C^3. Do not assume that the singularities of C_0 and C_1 are Newton non-degenerate.

Background

The paper introduces surface singularities in C^3 defined by g_j(z_1,z_2,z_3) = z_1^2 f_j(z_1,z_2,z_3) + h_j(z_2,z_3), where f_j is a reduced irreducible homogeneous polynomial of degree d defining a projective plane curve C_j ⊂ P^2, and h_j is a convenient homogeneous polynomial in z_2 and z_3 of degree d+3 satisfying Newton non-degeneracy on the face corresponding to z_1^2 f_j(0,z_2,z_3)+h_j(z_2,z_3) and the disjointness condition Sing(V_{P^2}(f_j)) ∩ V_{P^2}(h_j) = ∅.

Under the additional hypothesis that the singularities of C_0 and C_1 are Newton non-degenerate, the authors prove that the resulting germs g_0 and g_1 form a μ-Zariski pair. Motivated by Artal Bartolo’s results for superisolated singularities, they conjecture that when the Alexander polynomials of the curves C_0 and C_1 differ, the Jordan normal forms of the monodromies of g_0 and g_1 also differ, implying distinct embedded topologies of (V(g_0),0) and (V(g_1),0). They note that proving this would require extending Artal Bartolo’s theorem on superisolated singularities to this broader class of singularities.