An unusual family of supersingular curves of genus five in characteristic two

TL;DR

This paper constructs a family of supersingular genus 5 curves in characteristic two, highlighting their automorphisms, double cover structures, and explicit parametrization, advancing understanding of supersingular loci in algebraic geometry.

Contribution

It introduces a new explicit family of supersingular genus 5 curves in characteristic two with notable geometric features and automorphism properties.

Findings

Family has dimension matching the expected supersingular locus

Curves are non-hyperelliptic with non-trivial automorphisms

Each curve admits double covers over elliptic and genus-2 curves

Abstract

We construct a family of smooth supersingular curves of genus $5$ in characteristic $2$ with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus $5$, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-$2$ curve. We also provide an explicit parametrization of this family.