The 2-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms

TL;DR

This paper classifies genus 3 hyperelliptic curves with additional automorphisms by explicitly determining families with 2-Weierstrass points, using invariants of binary octavics, revealing that these families contain only genus 0 components.

Contribution

It explicitly determines families of genus 3 hyperelliptic curves with extra automorphisms that have 2-Weierstrass points, in terms of absolute invariants of binary octavics.

Findings

Families contain only genus 0 components

Explicit description using invariants of binary octavics

Classification for each automorphism group G with |G| > 2

Abstract

For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.