Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics

TL;DR

This paper introduces Z-splitting curves for double plane sextics, classifies lattice Zariski k-ples of simple sextics, and explores their specializations, advancing understanding of complex algebraic curves and their singularities.

Contribution

It defines Z-splitting curves for double plane sextics and classifies lattice types of these curves up to specialization, providing new insights into their structure.

Findings

Classified all lattice types of Z-splitting curves of degree ≤ 3

Introduced the notion of specialization of lattice types

Established a framework for lattice Zariski k-ples

Abstract

A simple sextic is a reduced complex projective plane curve of degree 6 with only simple singularities. We introduce a notion of Z-splitting curves for the double covering of the projective plane branching along a simple sextic, and investigate lattice Zariski k-ples of simple sextics by Z-splitting curves. We define specialization of lattice types, and classify all lattice types of Z-splitting curves of degree less than or equal to 3 up to specializations.