Explicit Constructions for Genus 3 Jacobians

TL;DR

This paper presents explicit algebraic constructions and algorithms for the Jacobian of genus 3 curves, including affine equations, group law descriptions, and embeddings into Grassmannian varieties.

Contribution

It introduces new explicit equations and algorithms for the Jacobian of genus 3 curves, extending Mumford's ideas for hyperelliptic cases.

Findings

Explicit equations for an affine open subset of the Jacobian.

Algorithms describing the group law on the Jacobian.

A construction embedding the Jacobian into a Grassmannian variety.

Abstract

Given a canonical genus three curve $X=\{F=0\}$, we construct, emulating Mumford discussion for hyperelliptic curves, a set of equations for an affine open subset of the jacobian $JX$. We give explicit algorithms describing the law group in $JX$. Finally we introduce a related construction by means of an imbedding of the open set previously described in a Grassmanian variety.