Let be an unramified double cover, where is geneus . Then has genus by the Riemann-Hurwitz formula. Now, encodes lots of information about the geometry of , especially with the additional data of the theta divisor. It turns out that for double covers, there’s an abelian variety that contains a lot of this data.
We, in fact, can describe this abelian variety in many ways. The simplest is probably that we have a pullback map , and we actually get a short exact sequence of abelian varieties , and is called the Prym variety associated to the cover.
Another description is that we can define a map , given by . Then, we look at the fiber over zero. The kernel here actually breaks up into two components. The component containing is isomorphic to , and the other is just a translation of .
So the first real theorem about Prym varieties is the following:
Wirtinger’s Theorem: The theta divisor on induces twice a principal polarization on . That is, .
So if we set the space of unramified double covers of curves of genus , and the space of principally polarized abelian varieties of dimension , then we have a map .
This leads to one of my favorite theorems
Theorem: The closure of the image of contains the Jacobians of genus .
To see this, we’ll use what are called Wirtinger covers. If , then is a stable curve of genus . We can then set . Then we’ll need to work out the Jacobians of and . For , a line bundle is the same as one on , along with a nonzero complex number, that is, a map identifying the fibers over and . So . Similarly, we have , and we get vertical maps given by multiplication, norm and tensor product. This then presents where is a finite group, and so the connected component of the identity is . So allowing the double cover to degenerate gives Jacobians!
Thus, the Prym map is closely tied to the problem of figuring out which abelian varieties are Jacobians (for more details, see my ICTP talk).