The last post was on the generalities of Abelian varieties, and constructing a map. This time, we’re going to do it for a specific one, and the maps involved will all be useful later. We start out with a finite morphism of curves.
There’s a very powerful theorem which lets us do many things, called the Relative Duality Theorem. I won’t be proving it, just stating it, and even if the statement is a bit beyond anything else we do, we only really need one consequence of it:
Theorem (Relative Duality) Let be a smooth morphism of smooth projective varieties of relative dimension . Let be a coherent sheaf on flat over and be a coherent sheaf on . Suppose that for some , commutes with base change for . Then there is a canonical isomorphism , where and is the th derived functor of the composition .
Now, if you unravel this for curves, as above, you’ll realize that so , so all the derived functors drop out of the formula. Fiddling with it an using the projection formula, you can derive that is isomorphic to . This is really the fact we want. That and the fact that the ramification divisor is a section of . We set , which is thus also and .
Now, we shift gears for a bit and define a map . This map only makes sense for curves. Take . It is isomorphic to for some divisor on . We then look at , which is a divisor on . We define . I’ll leave it as an exercise to check that this is well defined. But here’s the fun fact: for all ! This implies that , also by simple algebraic manipulation.
This will all be useful later on, but now, we must define the Prym variety associated to a ramified covering of curves. We have our morphism , and we further assume that is injective. Our ideal situation is that and , so that teh orthogonal to , which we’ll call , is the kernel of map , and that should be the Prym variety. However, this required that we make quite a lot of choices, and we want things to be as canonical as possible.
So instead, we set and , and take and , so that we have the Riemann Theta Divisors to work with. We also choose with . We get a map by , and we can apply the stuff from the last post. When the smoke clears, we find out that if we pull back along this map, its restriction to turns out to be , and we’ll denote by the restriction to . Then the pullback of the canonical section of gives an isomorphism , and we get the commuting diagram described last time, for the Prym variety of the cover.