On the genus of curves in the generic Prym variety (Q1329626)

Let \(\pi: \widetilde {C}\to C\) be an étale double covering of a smooth curve \(C\) of genus \(g+1\). The Prym variety \(P(\widetilde {C},C)\) associated to this covering is the connected component containing 0 of the kernel of the norm map \(\text{Nm}_ \pi: J(\widetilde {C})\to J(C)\). \(P(\widetilde {C}, C)\) is a principally polarized abelian variety of dimension \(g\) and the above construction defines the Prym map \[ P_{g+1}: {\mathcal R}_{g+1}=\{ \text{pairs \((\widetilde {C}, C)\) as above}\}\to {\mathcal A}_ g, \] where \({\mathcal A}_ g\) denotes the moduli space of principally polarized abelian varieties of dimension \(g\). The purpose of the paper under review is to give a lower bound for the genus of curves in \(P(\widetilde {C}, C)\). To be more precise, it is shown that for a generic Prym variety \(P\) of dimension \(g=2\) or \(\geq 4\) there is no nonconstant map \(D\to P\) from a smooth curve \(D\) of genus \(<2g -2\) into \(P\). The idea of the proof is to make use of a rigidity theorem [\textit{G. P. Pirola}, J. Reine Angew. Math. 431, 75-89 (1992; Zbl 0753.14040)] stating that under the above hypotheses for a generic \(P\) the only deformations of nonconstant maps \(D\to P\) are obtained by translations.