Geometry of irreducible plane quartics and their quadratic residue conics

TL;DR

This paper introduces quadratic residue curves mod irreducible quartic curves, explores their properties, and applies these concepts to dihedral covers and Zariski pairs, revealing new geometric relationships.

Contribution

It defines quadratic residue curves mod irreducible quartics and investigates their properties, leading to new insights into dihedral covers and Zariski pairs.

Findings

Defined quadratic residue curves mod quartics

Studied properties of quadratic residue conics

Constructed examples of Zariski pairs

Abstract

Let $D$ be an irreducible plane curve. In this article, we first introduce a notion of a quadratic residue curve mod $D$, and study quadratic residue concis $C$ mod an irreducible quartic curve $Q$. As an application, we study a dihedral cover of $P^2$ with branch locus $C+Q$ and give two examples of Zariski pairs as by-products.