On Prym varieties for the coverings of some singular plane curves

Abstract

Let k be a field of characteristic zero containing a primitive nth root of unity. Let $C^0_n$ be a singular plane curve of degree n over k admitting an order n automorphism, n nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\text{ Prym }(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\text{ Prym }(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb {Q}(\zeta _n+\zeta ^{-1}_n)$.