Prym varieties associated with \(n\)-cyclic coverings of a hyperelliptic curve (Q1416693)

Given a smooth, projective, hyperelliptic curve \(H\) of genus \(g\), for every \(\eta\) of order \(n\) in the Jacobian \(JH\), let \(f : C \rightarrow H\) be the \(n\)-cyclic covering associated to \(\eta\). The Prym variety \(P:= \text{Prym} (C/M)\) is given by the zero component of \(\operatorname{Ker} N_f\), \(N_f : JC \rightarrow JH\). \(P\) is an abelian subvariety of \(JC\); the main result in the paper shows that \(P\) is isomorphic to a product \(JC_0\times JC_1\) of jacobians of curves. The curves \(JC_0, JC_1\) can be found via a hyperelliptic structure \(C\rightarrow {\mathbb P}^1\) obtained from the one on \(H\); since \(\text{Gal} (C/{\mathbb P}^1) \cong D_n = \langle j, \sigma\rangle\), where \(j^2=\sigma^n=1\), there are involutions \(j_{\nu}=j\sigma ^\nu\), \(\nu = 0,\dots,n-1\) which define double ramified coverings \(f_\nu : C \rightarrow C_\nu := C/\langle j_\nu \rangle\). The jacobians \(JC_\nu\) are included in \(P\) and it turns out that \(P \cong JC_0\times JC_1\).