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In this paper we consider the Prym map for double coverings of curves ofgenus $g$ ramified at $r>0$ points. That is, the map associating to a doubleramified covering its Prym variety. The generic Torelli theorem states that thePrym map is generically injective as soon as the dimension of the space ofcoverings is less or equal to the dimension of the space of polarized abelianvarieties. We prove the generic injectivity of the Prym map in the cases ofdouble coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. Inthe 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 theintermediate Jacobians of cubic threefolds to conclude the generic injectivity.This completes the work of Marcucci and Pirola who proved this theorem for allthe other cases, except for the bielliptic case $g=1$ (solved later by Marcucciand the first author), and the case $g=3, r=4$ considered previously by Nagarajand Ramanan, and also by Bardelli, Ciliberto and Verra where the degree of themap is $3$.  The paper closes with an appendix by Alessandro Verra with an independentresult, the rationality of the moduli space of coverings with $g=2,r=6$, whoseproof is self-contained.

Generic injectivity of the Prym map for double ramified coverings