On degree three curves in $C^{(2)}$ with positive self-intersection

TL;DR

This paper investigates degree three algebraic curves within the symmetric square of a curve, focusing on those with positive self-intersection, using group actions to analyze their geometric properties.

Contribution

It introduces a novel approach to studying degree three curves with positive self-intersection via the action of spherical triangular groups.

Findings

Classification of degree three curves with positive self-intersection

Identification of conditions under which these curves exist

Insights into the geometric structure of such curves

Abstract

In this paper we study degree three curves in $C^{(2)}$ with positive self-intersection defined by the action of a spherical triangular group in an auxiliary curve.