On symmetric plane quartic curves

TL;DR

This paper investigates the geometric properties of smooth plane quartic curves with large automorphism groups, focusing on those invariant under specific group actions and analyzing their automorphism groups of order 48.

Contribution

It provides new insights into the symmetry groups of plane quartics, especially those with automorphism groups of order 48, and characterizes their geometric features.

Findings

Quartics invariant under elementary abelian groups of type [2,2,2] are characterized.

Smooth quartics with automorphism group of order 48 are studied in detail.

New geometric properties related to large automorphism groups are identified.

Abstract

In the present paper we study the geometry of plane quartics with large automorphism groups. We show results devoted to smooth plane quartics that are invariant under the action of the elementary abelian group of type $[2,2,2]$, and we study geometric properties of the smooth plane quartic having automorphism group of order $48$.