On Prym varieties for the coverings of some singular plane curves

TL;DR

This paper investigates Prym varieties associated with certain singular plane curves of degree n, providing explicit models and analyzing their factors, especially those with endomorphism rings containing specific number fields.

Contribution

It offers explicit models of algebraic curves related to Prym varieties of singular plane curves and identifies their simple factors with special endomorphism properties.

Findings

Explicit models of algebraic curves related to Prym varieties

Identification of simple factors with specific endomorphism rings

Analysis of Prym varieties for singular plane curves with automorphisms

Abstract

Let $k$ be a field of characteristic zero containing a primitive $n$-th 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 $\mbox{Prym}(\widetilde{C}_n/C_n)$ 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 $\mbox{Prym}(\widetilde{C}_n/C_n)$ 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 $\mathbb{Q}(\zeta_n+\zeta^{-1}_n)$.