On Prym varieties for the coverings of some singular plane curves (Q1755529)

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity and $C^{0}_{n}$ a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism and $n$ nodes as singularities. Consider its normalisation $C_{n}$ and the canonical double cover $\pi:\widetilde{C}_{n}\to C_{n}$ ramified precisely over the $2n$ preimages of the nodes. The Prym variety $\mathrm{Prym}(\widetilde{C}_{n}/C_{n})$ of this cover, defined as the complementary abelian variety to $\pi^{*}\mathrm{Jac}(C_{n})$ inside $\mathrm{Jac}(\widetilde{C}_{n})$, inherits a principal polarisation. \par In this paper, the authors study the isogenous decomposition of the Prym variety $\mathrm{Prym}(\widetilde{C}_{n}/C_{n})$ for a family of curves $C^{0}_{n}$ depending on certain parameters. Some of the factors in this decomposition turn out to be Prym varieties of lower genus curves of degree $n$, the endomorphism ring of which contains the totally real field $\mathbb{Q}(\zeta_{n}+\zeta_{n}^{-1})$. The main techniques in the proof involve the study of the automorphism group of the members of this family and their spaces of holomorphic differentials. In the last section, the computations in the case $n=5$ are carried out explicitly.