On the genus of curves in a Jacobian variety (Q2864591)

It was conjectured in [\textit{J. C. Naranjo} and \textit{G. P. Pirola}, Indag. Math., New Ser. 5, No. 1, 101--105 (1994; Zbl 0823.14033)] that a smooth curve \(D\) lying on a very generic Jacobian variety \(J(C)\) has to satisfy \(g(D)\geq 2g(C)-2\) or \(g(D)\leq g(C)\) if \(g(C)\geq 4\). Using a degeneration argument (borrowed from [\textit{F. Bardelli} and \textit{G. P. Pirola}, Invent. Math. 95, No. 2, 263--276 (1989; Zbl 0638.14025)]) and a careful analysis of what happens in the limit, the author proves that the conjecture holds in this stronger form: the genus of \(D\) has to satisfy \(g(D)\geq 2g(C)-2\) or \(g(D)=g(C)\).NEWLINENEWLINEThe equality case is discussed at the end of the paper : if \(g(C)=4\) or \(5\), there is no curve \(D\) on \(J(C)\) whose genus is \(g(D)=2g(C)-2\). The author finally conjectures that the lower bound should be replaced with \(g(D)\geq 2g(C)-1\).