Very interesting algebraic geometry seminar by Hang Xue today, about his paper “The height of a canonical point in the Jacobian of a genus 4 curve.”
How do you get a canonical point in the Jacobian of a genus 4 curve?
Well, that turns out to be kind of cool!
First of all, what does it mean for a point to be “canonical”? It should mean something like “functorial,” which should in turn mean something like “defined over the whole moduli space.” In other words: let K be the function field of M_4/C, and let X/K be the (restriction to generic moduli) of the universal genus-4 curve. Then we are asking for an element of the Mordell-Weil group Jac(X)(K). Except this Mordell-Weil group is trivial! This fact (or rather — this fact with 4 replaced by an arbitrary genus g) used to be called the Franchetta conjecture before it was proved by Beauville, Arbarello, and Cornalba.
So where does Xue’s point come from? It’s defined, not over K, but over a quadratic extension of K. The generic genus-4 curve embeds canonically in a quadric surface; the two families of lines aren’t defined over K, but over a quadratic extension K’, and passing to K’ you have made the quadric surface isomorphic to P^1 x P^1. Now there are two line bundles on P^1 x P^1 — pulling back each one to X gives you two degree-3 divisors on X, and their difference is a point on Jac(X)(K’). By construction, it is negated by conjugation over K.
Here’s another way to construct this point. Your generic genus-4 curve X has a degree-3 map to P^1 which is simply ramified at 12 points. So on X you have a ramification divisor D of degree 12. And then 2 K_X – D is your point on the Jacobian! When Hang explained this to me I said, wait, what happened to the quadratic extension? and he pointed out to me that the generic genus-4 curve actually has two trigonal maps to P^1; and these are defined over K’.
Anyway, Hang shows that this point P generates Jac(X)(K’) up to torsion, and moreover gives a nice formula for the Neron-Tate height of the restriction of P to any family of genus-4 curves over a 1-dimensional base.
Here is an idle question. Let d >= 1 be an integer. For each g, let K_g be the function field of M_g and J_g the Jacobian of the generic genus-g curve.
Are there only finitely many g such that J_g has a non-torsion point over some extension of K_g of degree at most d?
When d=1, this is the Franchetta conjecture.
I have no direct need to know the answer, but it seems a nice structural question about “universal Mordell-Weil groups.”