If $f \colon C \to C'$ is a dominant morphism of smooth projective curves, there is a norm map $f\_\ast = \mathrm{Nm} \colon JC \to JC'$ between their Jacobians, and we can consider the abelian subvariety $Z = (\ker \mathrm{Nm})^0$ and the polarization on $Z$ induced from $JC$. In two particularly interesting cases this polarization is twice a principal polarization $\Xi$, namely when $f$ has degree two and is either unramified or has two ramification points. In this case, one calls the ppav $(Z,\Xi)$ the Prym variety of $f$.


This construction works just as well in families, so it defines morphisms between some moduli spaces. If $R\_g$ denotes the moduli space of smooth genus $g$ curves $C'$ and an unramified connected double cover $C \to C'$, one gets a Prym map $R\_g \to A\_{g-1}$. Here $A\_g$ is the moduli space of dimension $g$ principally polarized abelian varieties. This map is rather well studied (e.g. the papers of Beauville, Donagi-Smith, Donagi and too many others to mention here). 


On the other hand, one could also let $R\_{g,2}$ be the moduli space of smooth genus $g$ curves with a double cover with two branch points. One gets in the same way a Prym map $R\_{g,2} \to A\_g$. But I have never seen any paper dealing with the properties of this map; maybe this is because of my incomplete knowledge of the literature. (One reason it may be less interesting is that unlike the ordinary Prym map in genus six, this one should never be generically finite for any $g$.)


Here are for instance some natural questions about this map: Is it dominant when $g \leq 4$? If yes, what is the structure of the generic fiber? Is it generically injective when $g \geq 5$? Does it extend to a compactification $\overline R\_{g,2}$ using admissible covers, either by mapping to the Satake compactification or to a toroidal compactification like the 2nd Voronoi? 


Is any of this known?



Has anyone studied the Prym map for double covers with two ramification points?
Hi, this is actually a part of my Ph.D. thesis. I am going to discuss it in 6 months. Here you can find a preprint of the work with my advisor <http://arxiv.org/abs/1010.4483>. It is not the final version, so there could be some minor mistakes.


We have proved that, with two ramification point, the Prym map is generically injective when g is greater or equal than 6 and that it is dominant for g=4. 


To complete the proof of the Generic Torelli Theorem, we have also studied a partial extension of the Prym map to admissible coverings. 


Anyway, I am still working on this topic and the thesis will contain other results about this.


