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 C0n be a singular plane curve of degree n over k admitting an order n automorphism, n nodes as the singularities, and Cn be its normalization. In this paper we study the factors of Prym variety Prym (C˜n/Cn) associated to the double cover C˜n of Cn 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 Prym (C˜n/Cn) as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in Jn so that the endomorphism rings contains the totally real field Q(ζn+ζ−1n).