(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”.)
Dogani
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?

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