(This post is incomplete and will later be proofread, filled in, etc.)
Today, new sunday routine, I search my “notes.org” file for lines containing the word “blog” and found this:
“brain dumps, of things I’ve learned recently and lots of fun facts (e.g., Pryms)”
So who is this blog’s target audience? Probably my own grad students (and, well, me at a later date). But non math people read this, and some senior math people too. What to do? Today for instance I’m going to write about Prym varieties. What can a non-math audience get out of a post like this? I’ll at least say that this encourages me to try to find pictures, facts, ect that I would not usually bother with.
Cool degenerations of curves
Another line says “blog post, cool degenerations of curves and more”, with a bunch of examples. Here’s a degenerate prym. (The top right curve is two copies of the left curve, but with “P of the first curve glued to Q of the second”.)
Two neat things about this picture
1) it is an etale double cover
2) The associated Prym variety is the Jacobian of the base curve
This second fact is key; it says that the Jacobian locus is in the closure of the Prym locus.
So what is a Prym variety?
The short answer is — it is (more or less) the only way that I know of to write down abelian varieties which are not Jacobians. I asked about this on Mathoverflow ages ago (examples of rational families of abelian varieties) and Prym varieties were (for me) the most managable example.
So how does one write down an abelian variety A that isn’t a Jacobian J? Well,
– A is dominated by a Jacobian (in particular, the Jacobian of some curve lying on A) so it makes sense to look for A’s in J’s.
– A map of curve C –> C’ induces a map of Jacobians J_C’ –> J_C, and the quotient A is an abelian variety that is often not a Jacobian
– This construction might not give you a principally polarized abelian variety.
– Geometric class field theory tells that two torsion points on J_C give etale double covers of C. This is (almost) the only case where one gets a PPVA.
Numerology of Prym’s
OK, the real reason I wanted to write this post is to collect all of the cool stuff I’ve learned about Pryms.
– Hyperelliptic begets hyperelliptic. In fact, this is an easy, explicit exercise (say, blay, in blay’s book):
– Trigonal begets tetragonal. Lies deeper. Dogani’s trigonal construction.
– Dimension counting. Call R_g the space of prym’s of dimension g. Then there is a map M_{g+1} –> R_g. It is a classical fact that this map is generically finite, with non-finite fibers coming from Dogani’s construction.
– Tetragonal usually begets a non-Jacobian.
– Tetragonal construction (characterize the tetragonal which do give Jacobians).
– Nodal degeneration trick. Keyword — Beauville admissible cover?
* Questions:
– Is there a good notion of Prym’s for graphs, or in tropical geometry? My first thought is no, but after Farbod S’s talk last week (reminding me of the emphasis on discrete graphs as models for metric graphs) maybe?
DZB's blog 
I just had a visitor with a strong interest in Prym varieties. There’s a fun paper of Nils Bruin about using Pryms to turn the number theory of genus 3 curves into that of genus two curves and eventually to get some crazy Hasse Principle exceptions.
Also, as you know, Pryms are generally quite hard to write down (as is any abelian variety), but last summer I thought about them and the Donagi(-Smith) construction for a little bit and there’s an instance where you can easily write them down. It’s actually outside of the usual etale double-covers domain of Prym varieties, but it works.
Any map from a genus two curve to a genus one curve has to be ramified at two (geometric) points. If the bottom curve is indeed an elliptic curve, we have to be able to write our genus two curve as
and our genus one curve as
This even works arithmetically, see papers of Everett Howe.
What then is the “Prym” for this map? Why it’s just the other curve that we map down to from our genus 2, namely
$
$
This comes up in a few fun places, such as elliptic curves of conductor 26, 38 and 58, where this double cover is in fact the modular map out of a Shimura curve.
… Hopefully that somewhat rambling comment fits in with the general “Here’s something cool!” feel of this blog.
Hi Jim! I like this example; when I get a chance to blog again I have another awesome example in the same spirit that I learned about a few weeks ago to blog about.