Zeta-function and $\mu^*$-Zariski pairs of surfaces

Abstract

A Zariski pair of surfaces is a pair of complex polynomial functions in $\mathbb{C}^3$ which is obtained from a classical Zariski pair of projective curves $f_0(z_1,z_2,z_3)=0$ and $f_1(z_1,z_2,z_3)=0$ of degree $d$ in $\mathbb{P}^2$ by adding a same term of the form $z_i^{d+m}$ ($m\geq 1$) to both $f_0$ and $f_1$ so that the corresponding affine surfaces of $\mathbb{C}^3$ -- defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ -- have an isolated singularity at the origin and the same zeta-function for the monodromy associated with their Milnor fibrations (so, in particular, $g_0$ and $g_1$ have the same Milnor number). In the present paper, we show that if $f_0$ and $f_1$ are "convenient" with respect to the coordinates $(z_1,z_2,z_3)$ and if the singularities of the curves $f_0=0$ and $f_1=0$ are Newton non-degenerate in some suitable local coordinates, then $(g_0,g_1)$ is a $\mu^*$-Zariski pair of surfaces, that is, a Zariski pair of surfaces whose polynomials $g_0$ and $g_1$ have the same Teissier's $\mu^*$-sequence but lie in different path-connected components of the $\mu^*$-constant stratum. To this end, we prove a new general formula that gives, under appropriate conditions, the Milnor number of functions of the above type, and we show (in a general setting) that two polynomials functions lying in the same path-connected component of the $\mu^*$-constant stratum can always be joined by a "piecewise complex-analytic path".