On the equivariant algebraic Jacobian for curves of genus two

TL;DR

This paper provides an algebraic framework for understanding the Jacobian of genus two curves, emphasizing SL2(k) symmetry and clarifying complex quadratic relations, with applications to the Kummer variety.

Contribution

It introduces an SL2(k)-equivariant algebraic approach to the Jacobian of genus two curves, clarifying Flynn's quadratic relations and extending to the Kummer variety.

Findings

Clarified the structure of Flynn's 72 quadratic relations.

Developed an SL2(k)-equivariant algebraic description.

Extended the framework to the Kummer variety.

Abstract

We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.