-ZARISKI PAIRS OF SURFACE SINGULARITIES

Pub Date : 2023-12-05 DOI:10.1017/nmj.2023.34

CHRISTOPHE EYRAL, MUTSUO OKA

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引用次数: 0

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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-zariski曲面奇点对

设$f_0$和$f_1$是三个复变量$z_1,z_2,z_3$的两个d次齐次多项式。我们证明了$g_0:=f_0+z_i^{d+m}$和$g_1:=f_1+z_i^{d+m}$定义的Lê-Yomdin曲面奇点具有相同的抽象拓扑，相同的单ζ函数，相同的$\mu ^*$ -不变量，但如果它们的投影切锥(分别由$f_0$和$f_1$定义)在$\mathbb {P}^2$中形成Zariski对曲线，则它们位于$\mu ^*$ -常数层的不同路径连通分量中。奇点是牛顿非简并的。在这种情况下，我们说$V(g_0):=g_0^{-1}(0)$和$V(g_1):=g_1^{-1}(0)$构成$\mu ^*$ -Zariski曲面奇点对。作为这样的一对是细菌$V(g_0)$和$V(g_1)$具有不同嵌入拓扑的必要条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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