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$\mu ^*$-ZARISKI PAIRS OF SURFACE SINGULARITIES
Published online by Cambridge University Press: 05 December 2023
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.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal
References
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