Arch. Math. 111 (2018), 621–631 c  2018 Springer Nature Switzerland AG 0003-889X/18/060621-11 published online August 25, 2018 https://doi.org/10.1007/s00013-018-1242-5 Archiv der Mathematik On the Prym map of cyclic coverings Herbert Lange and Angela Ortega Abstract. We work out the cases in which the Prym map is dominant and generically finite for ´ etale or totally ramified cyclic coverings. Mathematics Subject Classification. 14H40, 14H30. Keywords. Prym variety, Prym map. 1. Introduction. Let C be a smooth curve of genus g> 0 and f :  C → C be an ´ etale or totally ramified cyclic covering of degree d and ramification degree 2r, r ≥ 0 with r> 0 if g = 1. By totally ramified we mean that the ramification index at each ramification point is d. Note that by the Hurwitz formula the ramification degree is always even and the genus of  C is g(  C)= d(g − 1) + r +1. Let R g (d, r) denote the moduli space of such coverings. Then R g (d, r) is of dimension dim R g (d, r)=3g − 3+ 2r d − 1 . (1.1) Recall that the Prym variety P := P (f ) of a map [f :  C → C] ∈R g (d, r) is an abelian subvariety of J  C defined as the connected component containing the zero of the kernel of the norm map Nm f : J  C → JC,   i n i p i   →   i n i f (p i )  Since Nm f is a surjective homomorphism of groups, the dimension of P is given by dim P = g(  C) − g =(d − 1)(g − 1) + r. (1.2) The principal polarization of the Jacobian J  C of  C induces a polarization on P of some type which we denote by D.