The positive dimensional fibres of the Prym map (Q1919894)

Let \(C\) be an irreducible complex smooth curve of genus \(g\). Let \(\pi: \widetilde{C}\to C\) be a connected unramified double covering of \(C\). The Prym variety associated to the covering is, by definition, the component of the origin of the kernel of the norm map \(P(\widetilde{C},C)= \text{Ker(Nm}_\pi)^0\subset J\widetilde{C}\). It is a principally polarized abelian variety (p.p.a.v.) of dimension \(g(\widetilde{C})-g= g-1\). One defines the Prym map \[ \begin{aligned} P_g:{\mathcal R}_g &\to {\mathcal A}_{g-1},\\ (\widetilde{C}@>\pi>>C) &\mapsto P(\widetilde{C},C), \end{aligned} \] where \({\mathcal R}_g\) is the coarse moduli space of the coverings \(\pi\) as above and \({\mathcal A}_{g-1}\) stands for the coarse moduli space of p.p.a.v.'s of dimension \(g-1\). In this note we characterize the fibres of positive dimension of the Prym map. To state our theorem we need some notation: Let \({\mathcal {RB}}_g\) be the coarse moduli space of the unramified double coverings \(\pi:\widetilde{C}\to C\) such that \(C\) is a smooth bi-elliptic curve of genus \(g\). This variety has \([\frac{g-1}{2}]+2\) irreducible components \[ {\mathcal {RB}}_g=\Biggl( \bigcup_{t=0}^{[\frac{g-2}{2}]} {\mathcal {RB}}_{g,t}\Biggr)\cup {\mathcal {RB}}_g' \] [see \textit{J.-C. Naranjo}, J. Reine Angew. Math. 424, 47-106 (1992; Zbl 0733.14019)]. Theorem. Assume \(g\geq 13\). A fibre of \(P_g\) is positive dimensional at \((\widetilde{C},C)\) if and only if \(C\) is either hyperelliptic or \((\widetilde{C},C)\in \bigcup_{t\geq 1}{\mathcal {RB}}_{g,t}\).