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Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ on $P$ (where here $\equiv$ is numerical equivalence, and I am taking a principal polarization to mean an ample divisor whose space of global sections is 1-dimensional). $P$ is called a Prym-Tyurin variety of exponent $e$ for the curve $C$.

We have the Abel-Prym map associated to a point $p\in C$: $$\alpha_{p}:C\to P$$ $$q\mapsto i^\vee(q-p),$$ where $i^\vee$ is the dual of $i$ (remember that $P$ is principally polarized). This map can actually be defined in general for an abelian subvariety of $J$ by just taking the norm morphism, but this is the case I'm interested in.

For $e=1$, $P$ is just the Jacobian of $C$ and $\alpha_p$ is the usual Abel-Jacobi map, which is an embedding (this is a consequence of Matsusaka's Criterion).

For $e=2$, $\alpha_p$ is generally an embedding. Indeed, Welters showed that these Prym-Tyurin varieties can generally be constructed from a 2:1 étale cover of curves $f:C\to C'$, by taking $P$ to be the connected component of $$\ker(\mathcal{O}_C(D)\mapsto\mathcal{O}_{C'}(f(D)))$$ that contains 0 in $J$ (the Jacobian of $C$). If $C$ is not hyperelliptic then $\alpha_p$ is an embedding.

For $e=3$, Lange, Recillas and Rojas found in http://arxiv.org/pdf/math/0412103.pdf a family of Prym-Tyurin varieties of exponent $3$, and for large enough dimension (of the abelian varieties they study), they prove that $\alpha_p$ is an embedding.

My question is the following: Is it known if the Abel-Prym map is generally an embedding (or perhaps birational with its image)? Or is this maybe expected?

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