Nodal curves with a contact-conic and Zariski pairs

TL;DR

This paper investigates the splitting behavior of nodal plane curves with contact conics, introducing a new invariant called splitting type to distinguish their embedded topology and constructing examples with specific properties.

Contribution

It defines the splitting type of nodal curves with contact conics, provides a criterion for its determination, and constructs new Zariski-triples with prescribed splitting types.

Findings

Splitting type can distinguish embedded topology of plane curves.

A criterion for determining splitting type based on nodes and tangent points.

Construction of sextics and contact conics with specific splitting types leading to Zariski-triples.

Abstract

In this present paper, we study the splitting of nodal plane curves with respect to contact conics. We define the notion of splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.