GLOBAL PRYM-TORELLI FOR DOUBLE COVERINGS RAMIFIED IN AT LEAST SIX POINTS

We prove that the ramified Prym map P-g,P-r which sends a covering pi : D -> C ramified in r points to the Prym variety P(pi) := Ker(Nm(pi)) is an embedding for all r >= 6 and for all g(C) > 0. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that P-g,P-2 and P-g,P-4 have positive dimensional fibers.