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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?

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up vote 12 down vote accepted

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.

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Thanks a lot for this! –  Dan Petersen Jun 24 '11 at 4:33
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This is not as up to the minute as the beautifully detailed thesis linked in the answer above, but just a couple of historical comments.

Such a 2 point ramified double cover occurs from an unramified one when a loop shrinks to a point, which... let's see now, does intersect [I believe] the loop on which the double cover is based. Then I believe the induced double cover of the normalized curve is ramified at the 2 points over the singular point.

Vice versa, 2 point ramified double covers are a subcase of Beauville's "star - double covers" of curves with one node. E.g. 2 point ramified covers of genus 5 curves are a special case of Beauville's "star - double covers" of genus 6 curves with one node.

Anyway such ramified double covers have thus been studied by Wirtinger in the 19th century, and also explicitly by Fay in his book on theta functions, as well as more generally by Beauville.

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Thanks for the insight, this seems like a useful perspective. I guess another way of saying this is that you are identifying $R_{g,2}$ with a boundary stratum of the space $\overline{R}_{g+1}$ of admissible covers, and hence the two-pointed prym map with the restriction of the ordinary prym map in genus g+1. –  Dan Petersen Jun 24 '11 at 4:44
    
yes, Beauville's theory (and earlier partial steps by Masiewicki) shows that double covers with an even number of ramification points do yield a ppav once one identifies those points in pairs. But this does not help solve the ramified torelli problem since the torelli results for Beauville pryms are so generic, that the injectivity does not specialize. for covers that become singular so as to be still unramified however one can use induction on the normalization once that normalization becomes generic. Bob Friedman and I took this inductive approach to generic torelli for plane septics. –  roy smith Jun 24 '11 at 16:58
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