On the genus of curves in a Jacobian variety

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 small values of $g$ the second inequality can be further improved to $p>2g-2$.

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