As promised in the last post, I’m making another go at MaBloWriMo…maybe others will as well. I don’t know if I’m going to have a coherent topic over the course of this month, but I’ll be starting with abelian varieties associated to curves, so expect me to talk about generalizations of Prym varieties eventually (unless I get distracted by something else along the way). Today, the basic case: Jacobians!
We will be working exclusively over . Let be a curve. Then we have a short exact sequence of sheaves . This gives us a long exact sequence on cohomology, which simplifies to . The last four terms break off, and if we set , we have .
A fairly straightforward computation tells us that is a lattice in , and so is a complex torus. On a complex torus, we have that , and on , we have a pairing given by cup product. This is skew-symmetric, so gives .
And just as for the curve, we have a sequence and it can be shown that for some divisor . This isn’t uniquely defined, of course, but only up to , and as Jacobians and the curves they’re defined from have the same first cohomology groups, is defined up to a point of ! Also, any such divisor is ample, and defines an embedding of into of degree . So is an abelian variety, that is, a projective group variety.
As acts transitively on , and they are all isomorphic as abstract varieties (that is, without worrying about a natural group structure), is a torsor over . We can identify them all by picking a basepoint , and for any a divisor of degree , use the map by .
Tomorrow, we’ll talk about special loci in for .