Bleg: Independence of points in Picard group

For reasons discussed in my post below, I would like to construct a curve \overline{S} (over \mathbb{C}) and a finite map \overline{F}: S \to \mathbb{C}) with the following properties:

(1) The map \overline{F} is ramified somewhere. (This will be automatic if \overline{F} is not an isomorphism.) Let R \subset \overline{S} be the ramification locus and let S be \overline{S} \setminus R.

(2) The coordinate ring of S has no nontrivial units. This can be thought of as saying that R is not too large.

I think I have such a construction. But my argument that S has no units is basically that I need a bunch of elements of a Picard group to be linearly independent, and I can’t see any relations between them. Obviously, this needs some help! Details follow:

Let X be a generic genus 2 curve in \mathbb{P}^2. Let D = (d_1, d_2, d_3) be a generic divisor of degree 3 on X, and let \overline{S} = X \setminus D. A quick Riemmann-Roch computation shows that \mathcal{O}(D) has two linearly independent sections, so their ratio gives a map to \mathbb{CP}^1, which we can normalize so that D maps to \infty. So we get a map \overline{S} \to \mathbb{C}. The Riemann-Hurwitz formula shows that \overline{F} is ramified at 8 points, call them r_1, r_2, …, r_{8}. So we want to show that r_1, r_2, …, r_{8}, d_1, d_2, d_3 are linearly independent in \mathrm{Pic}(X).

We have the relation \sum r_i = 2 \sum d_i + K, where K is the canonical class and the equation takes place in \mathrm{Pic}(X) (exercise!). I can’t see any reason for any other relations to exist so I suspect that, assuming my choices are made generically, none will. Does anyone know how to prove this?

2 thoughts on “Bleg: Independence of points in Picard group

  1. Off the top of my head, if I really wanted to know this I guess I’d try to show that as you move around the space of trigonal genus-2 curves, the monodromy action on the ramification points by the full symmetric group on r_1, … r_8. Then any generic linear relation between the r_i and d_i would have to either involve only the d_i, or only the d_i and sum_j r_j, which can itself be expressed in terms of the d_i as you say.

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