-ZARISKI PAIRS OF SURFACE SINGULARITIES

Abstract

Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$ . We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$ -invariant, but lie in distinct path-connected components of the $\mu ^*$ -constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$ , respectively) make a Zariski pair of curves in $\mathbb {P}^2$ , the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$ -Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.