Characterizing Jacobians of algebraic curves with involution

TL;DR

This paper characterizes Jacobians of algebraic curves with involution and fixed points, linking their geometric properties to specific conditions involving Abelian subvarieties and the Kummer map within the context of Welter's trisecant conjecture.

Contribution

It provides two new characterizations of Jacobians with involution, connecting geometric conditions to the structure of Abelian subvarieties and the Kummer map, advancing understanding of the trisecant conjecture.

Findings

Jacobians with involution contain a shifted Abelian subvariety orthogonal to a specific vector under the Kummer map.

Two geometric characterizations of such Jacobians are established.

Results relate to particular cases of Welter's trisecant conjecture.

Abstract

We give two characterizations of Jacobians of curves with involution having fixed points in the framework of two particular cases of Welter's trisecant conjecture. The geometric form of each of these characterizations is the statement that such Jacobians are exactly those containing a shifted Abelian subvariety whose image under the Kummer map is orthogonal to an explicitly given vector.