A note on splitting numbers for Galois covers and $\pi_1$-equivalent Zariski $k$-plets

TL;DR

This paper introduces splitting numbers for subvarieties in smooth varieties under Galois covers, establishing their invariance and using them to classify plane curves topologically, leading to the construction of infinitely many -plets with identical fundamental groups.

Contribution

It defines splitting numbers for subvarieties in Galois covers and applies them to classify plane curves topologically, proving the existence of -plets with the same  fundamental group.

Findings

Splitting numbers are invariant under certain homeomorphisms.

A necessary and sufficient condition for topological equivalence of specific plane curves.

Existence of -plets of plane curves with identical  fundamental groups.

Abstract

In this paper, we introduce \textit{splitting numbers} of subvarieties in a smooth variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. By splitting numbers, we give a necessary and sufficient condition for two plane curves of type $(b,m)$ to be topologically equivalent as pairs of the complex projective plane and plane curves, where a plane curve of type $(b,m)$ is an arrangement of two smooth plane curves of degree $3$ and $b$ defined by I.~Shimada. Consequently, we prove that there are $\pi_1$-equivalent Zariski $k$-plets for any $k\geq2$.