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$\mu ^*$-ZARISKI PAIRS OF SURFACE SINGULARITIES
- Part of
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- Journal:
- Nagoya Mathematical Journal / Volume 254 / June 2024
- Published online by Cambridge University Press:
- 05 December 2023, pp. 488-497
- Print publication:
- June 2024
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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.
