Mult-Projective Models for Jacobian Varieties of Genus Two Curves

TL;DR

This paper develops a comprehensive multi-projective model for the Jacobian of genus two curves, providing explicit equations and a complete geometric description of the Picard group.

Contribution

It introduces multi-homogeneous equations and affine glue equations to model the Jacobian, completing a multi-projective and projective variety for genus two curves.

Findings

Explicit multi-homogeneous equations for Jacobians

Complete multi-projective model of the Jacobian variety

Geometric description of the Picard group

Abstract

A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimensional variety under the Segre imbedding.