On the genus of curves in a Jacobian variety

TL;DR

This paper proves bounds on the geometric genus of curves in very generic Jacobians, confirming a conjecture and providing improved inequalities for low dimensions, advancing understanding of the structure of these curves.

Contribution

It establishes new bounds on the geometric genus of curves in very generic Jacobians, confirming a conjecture and refining inequalities for small g.

Findings

If g>3, the genus p is either g or greater than 2g-3.

For small g, the bound improves to p>2g-2.

The results affirm a conjecture by Naranjo and Pirola.

Abstract

We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For low values of g the second inequality can be further improved to p>2g-2.