Abel-Prym maps for isotypical components of Jacobians

TL;DR

This paper studies the degree of Abel-Prym maps associated with isotypical components of Jacobians of complex curves, providing bounds and explicit computations that demonstrate their sharpness.

Contribution

It establishes bounds for the degree of Abel-Prym maps on isotypical components of Jacobians and computes explicit examples confirming these bounds are optimal.

Findings

Derived lower bounds for the degree of Abel-Prym maps.

Established upper bounds under certain conditions.

Computed explicit degrees in four specific cases.

Abstract

Let $C$ be a smooth non rational projective curve over the complex field $\mathbb{C}$. If $A$ is an abelian subvariety of the Jacobian $J(C)$, we consider the Abel-Prym map $\varphi_A : C \rightarrow A$ defined as the composition of the Abel map of $C$ with the norm map of $A$. The goal of this work is to investigate the degree of the map $\varphi_A$ in the case where $A$ is one of the components of an isotypical decomposition of $J(C)$. In this case we obtained a lower bound for $\mathrm{deg}(\varphi_A)$ and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of Abel-Prym maps in four cases. In particular, these examples show that both bounds are sharp.