Generic injectivity of the Prym map for double ramified coverings

Abstract

In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In the first case the proof is constructive and can be extended to the range $r\ge \max \{6,\frac 23(g+2) \}$. For (b) we study the fibre along the locus of the intermediate Jacobians of cubic threefolds to conclude the generic injectivity. This completes the work of Marcucci and Pirola who proved this theorem for all the other cases, except for the bielliptic case $g=1$ (solved later by Marcucci and the first author), and the case $g=3, r=4$ considered previously by Nagaraj and Ramanan, and also by Bardelli, Ciliberto and Verra where the degree of the map is $3$. The paper closes with an appendix by Alessandro Verra with an independent result, the rationality of the moduli space of coverings with $g=2,r=6$, whose proof is self-contained.