Zariski multiples associated with quartic curves

TL;DR

This paper studies special plane curves called Zariski multiples, showing their deformation types match their topological types and analyzing how the number of these types grows with the degree of the curves.

Contribution

It introduces a class of Zariski multiples associated with quartic curves and establishes a growth rate for their deformation types as the degree increases.

Findings

Deformation types equal homeomorphism types for these curves.

Number of deformation types grows as O(d^{62}) with degree d.

Provides a classification framework for Zariski multiples of quartic curves.

Abstract

We investigate Zariski multiples of plane curves $Z_1, \dots, Z_N$ such that each $Z_i$ is a union of a smooth quartic curve, some of its bitangents, and some of its 4-tangent conics. We show that, for plane curves of this type, the deformation types are equal to the homeomorphism types, and that the number of deformation types grows as $O(d^{62})$ when the degree $d$ of the plane curves tends to infinity.